{"id":540020,"date":"2021-09-25T08:04:21","date_gmt":"2021-09-25T08:04:21","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=540020"},"modified":"2022-12-26T09:23:34","modified_gmt":"2022-12-26T09:23:34","slug":"rd-sharma-solutions-for-class-12-maths-chapter-5-algebra-of-matrices","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/rd-sharma-solutions-for-class-12-maths-chapter-5-algebra-of-matrices\/","title":{"rendered":"RD Sharma Solutions for Class 12 Maths Chapter 5\u2013Algebra of Matrices"},"content":{"rendered":"\n

Class 12: Maths Chapter 5 solutions. Complete Class 12 Maths Chapter 5 Notes.<\/p>\n\n\n\n

RD Sharma Solutions for Class 12 Maths Chapter 5\u2013Algebra of Matrices<\/h2>\n\n\n\n

RD Sharma 12th Maths Chapter 5, Class 12 Maths Chapter 5 solutions<\/p>\n\n\n\n

Exercise 5.1 Page No: 5.6<\/p>\n\n\n\n

1. If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

If a matrix is of order m \u00d7 n elements, it has m n elements. So, if the matrix has 8 elements, we will find the ordered pairs m and n.<\/p>\n\n\n\n

m n = 8<\/p>\n\n\n\n

Then, ordered pairs m and n will be<\/p>\n\n\n\n

m \u00d7 n be (8 \u00d7 1),(1 \u00d7 8),(4 \u00d7 2),(2 \u00d7 4)<\/p>\n\n\n\n

Now, if it has 5 elements<\/p>\n\n\n\n

Possible orders are (5 \u00d7 1), (1 \u00d7 5).<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, Comparing with equation (1) and (2)<\/p>\n\n\n\n

a22<\/sub> = 4 and b21<\/sub> = \u2013 3<\/p>\n\n\n\n

a22<\/sub> + b21<\/sub> = 4 + (\u2013 3) = 1<\/p>\n\n\n\n

(ii)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, Comparing with equation (1) and (2)<\/p>\n\n\n\n

a11<\/sub> = 2, a22<\/sub> = 4, b11<\/sub> = 2, b22<\/sub> = 4<\/p>\n\n\n\n

a11<\/sub> b11<\/sub> + a22<\/sub> b22<\/sub> = 2 \u00d7 2 + 4 \u00d7 4 = 4 + 16 = 20<\/p>\n\n\n\n

3. Let A be a matrix of order 3 \u00d7 4. If R1<\/sub> denotes the first row of A and C2<\/sub> denotes its second column, then determine the orders of matrices R1<\/sub> and C2<\/sub>.<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given A be a matrix of order 3 \u00d7 4.<\/p>\n\n\n\n

So, A = [ai j<\/sub>] 3\u00d74<\/sub><\/p>\n\n\n\n

R1<\/sub> = first row of A = [a11<\/sub>, a12<\/sub>, a13<\/sub>, a14<\/sub>]<\/p>\n\n\n\n

So, order of matrix R1<\/sub> = 1 \u00d7 4<\/p>\n\n\n\n

C2<\/sub> = second column of<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore order of C2<\/sub> = 3 \u00d7 1<\/p>\n\n\n\n

4. Construct a 2 \u00d73 matrix A = [aj j<\/sub>] whose elements aj j<\/sub> are given by:<\/strong><\/p>\n\n\n\n

(i) ai j<\/sub> = i \u00d7 j<\/strong><\/p>\n\n\n\n

(ii) ai j <\/sub>= 2i \u2013 j<\/strong><\/p>\n\n\n\n

(iii) ai j <\/sub>= i + j<\/strong><\/p>\n\n\n\n

(iv) ai j<\/sub> = (i + j)2<\/sup>\/2<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given ai j<\/sub> = i \u00d7 j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2 \u00d7 3<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 3 matrix are[a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub>]<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 1 \u00d7 1 = 1<\/p>\n\n\n\n

a12<\/sub> = 1 \u00d7 2 = 2<\/p>\n\n\n\n

a13<\/sub> = 1 \u00d7 3 = 3<\/p>\n\n\n\n

a21<\/sub> = 2 \u00d7 1 = 2<\/p>\n\n\n\n

a22<\/sub> = 2 \u00d7 2 = 4<\/p>\n\n\n\n

a23<\/sub> = 2 \u00d7 3 = 6<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Given ai j <\/sub>= 2i \u2013 j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 2 \u00d7 1 \u2013 1 = 2 \u2013 1 = 1<\/p>\n\n\n\n

a12<\/sub> = 2 \u00d7 1 \u2013 2 = 2 \u2013 2 = 0<\/p>\n\n\n\n

a13<\/sub> = 2 \u00d7 1 \u2013 3 = 2 \u2013 3 = \u2013 1<\/p>\n\n\n\n

a21<\/sub> = 2 \u00d7 2 \u2013 1 = 4 \u2013 1 = 3<\/p>\n\n\n\n

a22<\/sub> = 2 \u00d7 2 \u2013 2 = 4 \u2013 2 = 2<\/p>\n\n\n\n

a23<\/sub> = 2 \u00d7 2 \u2013 3 = 4 \u2013 3 = 1<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iii) Given ai j <\/sub>= i + j<\/p>\n\n\n\n

Let A = [a i j<\/sub>] 2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 1 + 1 = 2<\/p>\n\n\n\n

a12<\/sub> = 1 + 2 = 3<\/p>\n\n\n\n

a13<\/sub> = 1 + 3 = 4<\/p>\n\n\n\n

a21<\/sub> = 2 + 1 = 3<\/p>\n\n\n\n

a22<\/sub> = 2 + 2 = 4<\/p>\n\n\n\n

a23<\/sub> = 2 + 3 = 5<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

(iv) Given ai j<\/sub> = (i + j)2<\/sup>\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a13<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a23<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

5. Construct a 2 \u00d7 2 matrix A = [ai j<\/sub>] whose elements ai j<\/sub> are given by:<\/strong><\/p>\n\n\n\n

(i) (i + j)2 <\/sup>\/2<\/strong><\/p>\n\n\n\n

(ii) ai j<\/sub> = (i \u2013 j)2 <\/sup>\/2<\/strong><\/p>\n\n\n\n

(iii) ai j<\/sub> = (i \u2013 2j)2 <\/sup>\/2<\/strong><\/p>\n\n\n\n

(iv) ai j<\/sub> = (2i + j)2 <\/sup>\/2<\/strong><\/p>\n\n\n\n

(v) ai j<\/sub> = |2i \u2013 3j|\/2<\/strong><\/p>\n\n\n\n

(vi) ai j<\/sub> = |-3i + j|\/2<\/strong><\/p>\n\n\n\n

(vii) ai j<\/sub> = e2ix<\/sup> sin x j<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given (i + j)2 <\/sup>\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 2 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(ii) Given ai j<\/sub> = (i \u2013 j)2 <\/sup>\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 2 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(iii) Given ai j<\/sub> = (i \u2013 2j)2 <\/sup>\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 2 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(iv) Given ai j<\/sub> = (2i + j)2 <\/sup>\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 2 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(v) Given ai j<\/sub> = |2i \u2013 3j|\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2\u00d72 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(vi) Given ai j<\/sub> = |-3i + j|\/2<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 2 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(vii) Given ai j<\/sub> = e2ix<\/sup> sin x j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d72<\/sub><\/p>\n\n\n\n

So, the elements in a 2 \u00d7 2 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a21<\/sub>, a22<\/sub>,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

6. Construct a 3\u00d74 matrix A = [ai j<\/sub>] whose elements ai j<\/sub> are given by:
(i) ai j<\/sub> = i + j<\/strong><\/p>\n\n\n\n

(ii) ai j<\/sub> = i \u2013 j<\/strong><\/p>\n\n\n\n

(iii) ai j<\/sub> = 2i<\/strong><\/p>\n\n\n\n

(iv) ai j<\/sub> = j<\/strong><\/p>\n\n\n\n

(v) ai j<\/sub> = \u00bd |-3i + j|<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given ai j<\/sub> = i + j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 3 \u00d7 4 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a14,<\/sub> a21<\/sub>, a22<\/sub>, a23<\/sub>, a24<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a34<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 1 + 1 = 2<\/p>\n\n\n\n

a12<\/sub> = 1 + 2 = 3<\/p>\n\n\n\n

a13<\/sub> = 1 + 3 = 4<\/p>\n\n\n\n

a14<\/sub> = 1 + 4 = 5<\/p>\n\n\n\n

a21<\/sub> = 2 + 1 = 3<\/p>\n\n\n\n

a22<\/sub> = 2 + 2 = 4<\/p>\n\n\n\n

a23<\/sub> = 2 + 3 = 5<\/p>\n\n\n\n

a24 =<\/sub> 2 + 4 = 6<\/p>\n\n\n\n

a31<\/sub> = 3 + 1 = 4<\/p>\n\n\n\n

a32<\/sub> = 3 + 2 = 5<\/p>\n\n\n\n

a33<\/sub> = 3 + 3 = 6<\/p>\n\n\n\n

a34<\/sub> = 3 + 4 = 7<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(ii) Given ai j<\/sub> = i \u2013 j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 3\u00d74 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a14,<\/sub> a21<\/sub>, a22<\/sub>, a23<\/sub>, a24<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a34<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 1 \u2013 1 = 0<\/p>\n\n\n\n

a12<\/sub> = 1 \u2013 2 = \u2013 1<\/p>\n\n\n\n

a13<\/sub> = 1 \u2013 3 = \u2013 2<\/p>\n\n\n\n

a14<\/sub> = 1 \u2013 4 = \u2013 3<\/p>\n\n\n\n

a21<\/sub> = 2 \u2013 1 = 1<\/p>\n\n\n\n

a22<\/sub> = 2 \u2013 2 = 0<\/p>\n\n\n\n

a23<\/sub> = 2 \u2013 3 = \u2013 1<\/p>\n\n\n\n

a24 =<\/sub> 2 \u2013 4 = \u2013 2<\/p>\n\n\n\n

a31<\/sub> = 3 \u2013 1 = 2<\/p>\n\n\n\n

a32<\/sub> = 3 \u2013 2 = 1<\/p>\n\n\n\n

a33<\/sub> = 3 \u2013 3 = 0<\/p>\n\n\n\n

a34<\/sub> = 3 \u2013 4 = \u2013 1<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(iii) Given ai j<\/sub> = 2i<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 3\u00d74 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a14,<\/sub> a21<\/sub>, a22<\/sub>, a23<\/sub>, a24<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a34<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 2\u00d71 = 2<\/p>\n\n\n\n

a12<\/sub> = 2\u00d71 = 2<\/p>\n\n\n\n

a13<\/sub> = 2\u00d71 = 2<\/p>\n\n\n\n

a14<\/sub> = 2\u00d71 = 2<\/p>\n\n\n\n

a21<\/sub> = 2\u00d72 = 4<\/p>\n\n\n\n

a22<\/sub> = 2\u00d72 = 4<\/p>\n\n\n\n

a23<\/sub> = 2\u00d72 = 4<\/p>\n\n\n\n

a24 =<\/sub> 2\u00d72 = 4<\/p>\n\n\n\n

a31<\/sub> = 2\u00d73 = 6<\/p>\n\n\n\n

a32<\/sub> = 2\u00d73 = 6<\/p>\n\n\n\n

a33<\/sub> = 2\u00d73 = 6<\/p>\n\n\n\n

a34<\/sub> = 2\u00d73 = 6<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(iv) Given ai j<\/sub> = j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 3\u00d74 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a14,<\/sub> a21<\/sub>, a22<\/sub>, a23<\/sub>, a24<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a34<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 1<\/p>\n\n\n\n

a12<\/sub> = 2<\/p>\n\n\n\n

a13<\/sub> = 3<\/p>\n\n\n\n

a14<\/sub> = 4<\/p>\n\n\n\n

a21<\/sub> = 1<\/p>\n\n\n\n

a22<\/sub> = 2<\/p>\n\n\n\n

a23<\/sub> = 3<\/p>\n\n\n\n

a24 =<\/sub> 4<\/p>\n\n\n\n

a31<\/sub> = 1<\/p>\n\n\n\n

a32<\/sub> = 2<\/p>\n\n\n\n

a33<\/sub> = 3<\/p>\n\n\n\n

a34<\/sub> = 4<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(vi) Given ai j<\/sub> = \u00bd |-3i + j|<\/p>\n\n\n\n

Let A = [ai j<\/sub>]2\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 3\u00d74 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a14,<\/sub> a21<\/sub>, a22<\/sub>, a23<\/sub>, a24<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a34<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a13<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a14<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a23<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a24 =<\/sub>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a31<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a32<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a33<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a34<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Multiplying by negative sign we get,<\/p>\n\n\n\n

7. Construct a 4 \u00d7 3 matrix A = [ai j<\/sub>] whose elements ai j<\/sub> are given by:<\/strong><\/p>\n\n\n\n

(i) ai j<\/sub> = 2i + i\/j<\/strong><\/p>\n\n\n\n

(ii) ai j<\/sub> = (i \u2013 j)\/ (i + j)<\/strong><\/p>\n\n\n\n

(iii) ai j<\/sub> = i<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given ai j<\/sub> = 2i + i\/j<\/p>\n\n\n\n

Let A = [ai j<\/sub>]4\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 4 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a41,<\/sub> a42,<\/sub> a43<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a13<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a23<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a31<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a32<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a33<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a41<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a42<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a43<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(ii) Given ai j<\/sub> = (i \u2013 j)\/ (i + j)<\/p>\n\n\n\n

Let A = [ai j<\/sub>]4\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 4 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a41,<\/sub> a42,<\/sub> a43<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a12<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a13<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a21<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a22<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a23<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a31<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a32<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a33<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a41<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a42<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a43<\/sub> =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(iii) Given ai j<\/sub> = i<\/p>\n\n\n\n

Let A = [ai j<\/sub>]4\u00d73<\/sub><\/p>\n\n\n\n

So, the elements in a 4 \u00d7 3 matrix are<\/p>\n\n\n\n

a11<\/sub>, a12<\/sub>, a13<\/sub>, a21<\/sub>, a22<\/sub>, a23<\/sub>, a31, <\/sub>a32, <\/sub>a33, <\/sub>a41,<\/sub> a42,<\/sub> a43<\/sub><\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a11<\/sub> = 1<\/p>\n\n\n\n

a12<\/sub> = 1<\/p>\n\n\n\n

a13<\/sub> = 1<\/p>\n\n\n\n

a21<\/sub> = 2<\/p>\n\n\n\n

a22<\/sub> = 2<\/p>\n\n\n\n

a23<\/sub> = 2<\/p>\n\n\n\n

a31<\/sub> = 3<\/p>\n\n\n\n

a32<\/sub> = 3<\/p>\n\n\n\n

a33<\/sub> = 3<\/p>\n\n\n\n

a41<\/sub> = 4<\/p>\n\n\n\n

a42<\/sub> = 4<\/p>\n\n\n\n

a43<\/sub> = 4<\/p>\n\n\n\n

Substituting these values in matrix A we get,<\/p>\n\n\n\n

A =
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

8. Find x, y, a and b if<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Given that two matrices are equal.<\/p>\n\n\n\n

We know that if two matrices are equal then the elements of each matrices are also equal.<\/p>\n\n\n\n

Therefore by equating them we get,<\/p>\n\n\n\n

3x + 4y = 2 \u2026\u2026 (1)<\/p>\n\n\n\n

x \u2013 2y = 4 \u2026\u2026 (2)<\/p>\n\n\n\n

a + b = 5 \u2026\u2026 (3)<\/p>\n\n\n\n

2a \u2013 b = \u2013 5 \u2026\u2026 (4)<\/p>\n\n\n\n

Multiplying equation (2) by 2 and adding to equation (1), we get<\/p>\n\n\n\n

3x + 4y + 2x \u2013 4y = 2 + 8<\/p>\n\n\n\n

\u21d2 5x = 10<\/p>\n\n\n\n

\u21d2 x = 2<\/p>\n\n\n\n

Now, substituting the value of x in equation (1)<\/p>\n\n\n\n

3 \u00d7 2 + 4y = 2<\/p>\n\n\n\n

\u21d2 6 + 4y = 2<\/p>\n\n\n\n

\u21d2 4y = 2 \u2013 6<\/p>\n\n\n\n

\u21d2 4y = \u2013 4<\/p>\n\n\n\n

\u21d2 y = \u2013 1<\/p>\n\n\n\n

Now by adding equation (3) and (4)<\/p>\n\n\n\n

a + b + 2a \u2013 b = 5 + (\u2013 5)<\/p>\n\n\n\n

\u21d2 3a = 5 \u2013 5 = 0<\/p>\n\n\n\n

\u21d2 a = 0<\/p>\n\n\n\n

Now, again by substituting the value of a in equation (3), we get<\/p>\n\n\n\n

0 + b = 5<\/p>\n\n\n\n

\u21d2 b = 5<\/p>\n\n\n\n

\u2234 a = 0, b = 5, x = 2 and y = \u2013 1<\/p>\n\n\n\n

9. Find x, y, a and b if<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

We know that if two matrices are equal then the elements of each matrices are also equal.<\/p>\n\n\n\n

Given that two matrices are equal.<\/p>\n\n\n\n

Therefore by equating them we get,<\/p>\n\n\n\n

2a + b = 4 \u2026\u2026 (1)<\/p>\n\n\n\n

And a \u2013 2b = \u2013 3 \u2026\u2026 (2)<\/p>\n\n\n\n

And 5c \u2013 d = 11 \u2026\u2026 (3)<\/p>\n\n\n\n

4c + 3d = 24 \u2026\u2026 (4)<\/p>\n\n\n\n

Multiplying equation (1) by 2 and adding to equation (2)<\/p>\n\n\n\n

4a + 2b + a \u2013 2b = 8 \u2013 3<\/p>\n\n\n\n

\u21d2 5a = 5<\/p>\n\n\n\n

\u21d2 a = 1<\/p>\n\n\n\n

Now, substituting the value of a in equation (1)<\/p>\n\n\n\n

2 \u00d7 1 + b = 4<\/p>\n\n\n\n

\u21d2 2 + b = 4<\/p>\n\n\n\n

\u21d2 b = 4 \u2013 2<\/p>\n\n\n\n

\u21d2 b = 2<\/p>\n\n\n\n

Multiplying equation (3) by 3 and adding to equation (4)<\/p>\n\n\n\n

15c \u2013 3d + 4c + 3d = 33 + 24<\/p>\n\n\n\n

\u21d2 19c = 57<\/p>\n\n\n\n

\u21d2 c = 3<\/p>\n\n\n\n

Now, substituting the value of c in equation (4)<\/p>\n\n\n\n

4 \u00d7 3 + 3d = 24<\/p>\n\n\n\n

\u21d2 12 + 3d = 24<\/p>\n\n\n\n

\u21d2 3d = 24 \u2013 12<\/p>\n\n\n\n

\u21d2 3d = 12<\/p>\n\n\n\n

\u21d2 d = 4<\/p>\n\n\n\n

\u2234 a = 1, b = 2, c = 3 and d = 4<\/p>\n\n\n\n

10. Find the values of a, b, c and d from the following equations:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

We know that if two matrices are equal then the elements of each matrices are also equal.<\/p>\n\n\n\n

Given that two matrices are equal.<\/p>\n\n\n\n

Therefore by equating them we get,<\/p>\n\n\n\n

2a + b = 4 \u2026\u2026 (1)<\/p>\n\n\n\n

And a \u2013 2b = \u2013 3 \u2026\u2026 (2)<\/p>\n\n\n\n

And 5c \u2013 d = 11 \u2026\u2026 (3)<\/p>\n\n\n\n

4c + 3d = 24 \u2026\u2026 (4)<\/p>\n\n\n\n

Multiplying equation (1) by 2 and adding to equation (2)<\/p>\n\n\n\n

4a + 2b + a \u2013 2b = 8 \u2013 3<\/p>\n\n\n\n

\u21d2 5a = 5<\/p>\n\n\n\n

\u21d2 a = 1<\/p>\n\n\n\n

Now, substituting the value of a in equation (1)<\/p>\n\n\n\n

2 \u00d7 1 + b = 4<\/p>\n\n\n\n

\u21d2 2 + b = 4<\/p>\n\n\n\n

\u21d2 b = 4 \u2013 2<\/p>\n\n\n\n

\u21d2 b = 2<\/p>\n\n\n\n

Multiplying equation (3) by 3 and adding to equation (4)<\/p>\n\n\n\n

15c \u2013 3d + 4c + 3d = 33 + 24<\/p>\n\n\n\n

\u21d2 19c = 57<\/p>\n\n\n\n

\u21d2 c = 3<\/p>\n\n\n\n

Now, substituting the value of c in equation (4)<\/p>\n\n\n\n

4 \u00d7 3 + 3d = 24<\/p>\n\n\n\n

\u21d2 12 + 3d = 24<\/p>\n\n\n\n

\u21d2 3d = 24 \u2013 12<\/p>\n\n\n\n

\u21d2 3d = 12<\/p>\n\n\n\n

\u21d2 d = 4<\/p>\n\n\n\n

\u2234 a = 1, b = 2, c = 3 and d = 4<\/p>\n\n\n\n

\n\n\n\n

Exercise 5.2 Page No: 5.18<\/p>\n\n\n\n

1. Compute the following sums:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Corresponding elements of two matrices should be added<\/p>\n\n\n\n

Therefore, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Find each of the following:<\/strong><\/p>\n\n\n\n

(i) 2A \u2013 3B<\/strong><\/p>\n\n\n\n

(ii) B \u2013 4C<\/strong><\/p>\n\n\n\n

(iii) 3A \u2013 C<\/strong><\/p>\n\n\n\n

(iv) 3A \u2013 2B + 3C<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have to compute 2A<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by computing 3B we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by we have to compute 2A \u2013 3B we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Given
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have to compute 4C,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have to compute 3A,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iv) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have to compute 3A<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to compute 2B<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

By computing 3C we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(i) A + B and B + C<\/strong><\/p>\n\n\n\n

(ii) 2B + 3A and 3C \u2013 4B<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Consider A + B,<\/p>\n\n\n\n

A + B is not possible because matrix A is an order of 2 x 2 and Matrix B is an order of 2 x 3, so the Sum of the matrix is only possible when their order is same.<\/p>\n\n\n\n

Now consider B + C<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Consider 2B + 3A<\/p>\n\n\n\n

2B + 3A also does not exist because the order of matrix B and matrix A is different, so we cannot find the sum of these matrix.<\/p>\n\n\n\n

Now consider 3C \u2013 4B,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to compute 2A \u2013 3B + 4C<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

5. If A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4), find<\/strong><\/p>\n\n\n\n

(i) A \u2013 2B<\/strong><\/p>\n\n\n\n

(ii) B + C \u2013 2A<\/strong><\/p>\n\n\n\n

(iii) 2A + 3B \u2013 5C<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Given A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4)<\/p>\n\n\n\n

We have to find B + C \u2013 2A<\/p>\n\n\n\n

Here,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to compute B + C \u2013 2A<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iii) Given A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4)<\/p>\n\n\n\n

Now we have to find 2A + 3B \u2013 5C<\/p>\n\n\n\n

Here,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now consider 2A + 3B \u2013 5C<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

6. Given the matrices<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Verify that (A + B) + C = A + (B + C)<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to verify (A + B) + C = A + (B + C)<\/p>\n\n\n\n

First consider LHS, (A + B) + C,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now consider RHS, that is A + (B + C)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore LHS = RHS<\/p>\n\n\n\n

Hence (A + B) + C = A + (B + C)<\/p>\n\n\n\n

7. Find the matrices X and Y,<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by simplifying we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by simplifying we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by transposing, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now by multiplying equation (1) and (2) we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now by adding equation (2) and (3) we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by substituting X in equation (2) we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Consider<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, again consider<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

And<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\n\n\n\n

Exercise 5.3 Page No: 5.41<\/p>\n\n\n\n

1. Compute the indicated products:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Consider<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

On simplification we get,<\/p>\n\n\n\n

\"RD<\/figure>\n\n\n\n

(ii) Consider<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

On simplification we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iii) Consider<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

On simplification we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

2. Show that AB \u2260 BA in each of the following cases:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that<\/p>\n\n\n\n

AB \u2260 BA<\/p>\n\n\n\n

(ii) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that<\/p>\n\n\n\n

AB \u2260 BA<\/p>\n\n\n\n

(iii) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that<\/p>\n\n\n\n

AB \u2260 BA<\/p>\n\n\n\n

3. Compute the products AB and BA whichever exists in each of the following cases:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

BA does not exist<\/p>\n\n\n\n

Because the number of columns in B is greater than the rows in A<\/p>\n\n\n\n

(ii) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iii) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

AB = [0 + (-1) + 6 + 6]<\/p>\n\n\n\n

AB = 11<\/p>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iv) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

4. Show that AB \u2260 BA in each of the following cases:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that<\/p>\n\n\n\n

AB \u2260 BA<\/p>\n\n\n\n

(ii) Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2) it is clear that,<\/p>\n\n\n\n

AB \u2260 BA<\/p>\n\n\n\n

5. Evaluate the following:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have to add first two matrix,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

On simplifying, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have to multiply first two given matrix,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

= 82<\/p>\n\n\n\n

(iii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

First we have subtract the matrix which is inside the bracket,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

We know that,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again we know that,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

We have,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, from equation (1), (2), (3) and (4), it is clear that A2 <\/sup>= B2<\/sup>= C2<\/sup>= I2<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to find,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

We know that,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Again we have,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2) AB = BA = 03\u00d73<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

From equation (1) and (2) AB = BA = 03\u00d73<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore AB = A<\/p>\n\n\n\n

Again consider, BA we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Hence BA = B<\/p>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now again consider, B2<\/sup><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by subtracting equation (2) from equation (1) we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

16. For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A (BC)<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now consider RHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that (AB) C = A (BC)<\/p>\n\n\n\n

(ii) Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider the LHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now consider RHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that (AB) C = A (BC)<\/p>\n\n\n\n

17. For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC.<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider LHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now consider RHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From equation (1) and (2), it is clear that A (B + C) = AB + AC<\/p>\n\n\n\n

(ii) Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider the LHS<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now consider RHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider the LHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now consider RHS<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

From the above equations LHS = RHS<\/p>\n\n\n\n

Therefore, A (B \u2013 C) = AB \u2013 AC.<\/p>\n\n\n\n

19. Compute the elements a43<\/sub> and a22<\/sub> of the matrix:<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

From the above matrix, a43<\/sub> = 8and a22<\/sub> = 0<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, consider the RHS<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Therefore, A3<\/sup> = p I + q A + rA2<\/sup><\/p>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

21. If \u03c9 is a complex cube root of unity, show that<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"RD<\/figure>\n\n\n\n

It is also given that \u03c9 is a complex cube root of unity,<\/p>\n\n\n\n

Consider the LHS,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

We know that 1 + \u03c9 + \u03c92<\/sup> = 0 and \u03c93<\/sup> = 1<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now by simplifying we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Again by substituting 1 + \u03c9 + \u03c92<\/sup> = 0 and \u03c93<\/sup> = 1 in above matrix we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore LHS = RHS<\/p>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider A2<\/sup><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore A2<\/sup> = A<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider A2<\/sup>,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Hence A2<\/sup> = I3<\/sub><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

= [2x + 1 + 2 + x + 3] = 0<\/p>\n\n\n\n

= [3x + 6] = 0<\/p>\n\n\n\n

= 3x = -6<\/p>\n\n\n\n

x = -6\/3<\/p>\n\n\n\n

x = -2<\/p>\n\n\n\n

(ii) Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

On comparing the above matrix we get,<\/p>\n\n\n\n

x = 13<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

\u21d2 [(2x + 4) x + 4 (x + 2) \u2013 1(2x + 4)] = 0<\/p>\n\n\n\n

\u21d2 2x2<\/sup> + 4x + 4x + 8 \u2013 2x \u2013 4 = 0<\/p>\n\n\n\n

\u21d2 2x2<\/sup> + 6x + 4 = 0<\/p>\n\n\n\n

\u21d2 2x2<\/sup> + 2x + 4x + 4 = 0<\/p>\n\n\n\n

\u21d2 2x (x + 1) + 4 (x + 1) = 0<\/p>\n\n\n\n

\u21d2 (x + 1) (2x + 4) = 0<\/p>\n\n\n\n

\u21d2 x = -1 or x = -2<\/p>\n\n\n\n

Hence, x = -1 or x = -2<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

By multiplying we get,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to prove A2<\/sup> \u2013 A + 2 I = 0<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Hence the proof.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

I is identity matrix, so<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Also given,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, we have to find A2<\/sup>, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now, we will find the matrix for 8A, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Substitute corresponding values from eqn (i) and (ii), we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal<\/p>\n\n\n\n

Hence,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Therefore, the value of k is 7<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

To show that f (A) = 0<\/p>\n\n\n\n

Substitute x = A in f(x), we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

I is identity matrix, so<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, we will find the matrix for A2<\/sup>, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now, we will find the matrix for 2A, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Substitute corresponding values from eqn (ii) and (iii) in eqn (i), we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Hence Proved<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

So<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now, we will find the matrix for A2<\/sup>, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now, we will find the matrix for \u03bb A, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

But given, A2<\/sup> = \u03bb A + \u03bc I<\/p>\n\n\n\n

Substitute corresponding values from equation (i) and (ii), we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal<\/p>\n\n\n\n

Hence, \u03bb + 0 = 4 \u21d2 \u03bb = 4<\/p>\n\n\n\n

And also, 2\u03bb + \u03bc = 7<\/p>\n\n\n\n

Substituting the obtained value of \u03bb in the above equation, we get<\/p>\n\n\n\n

2(4) + \u03bc = 7 \u21d2 8 + \u03bc = 7 \u21d2 \u03bc = \u2013 1<\/p>\n\n\n\n

Therefore, the value of \u03bb and \u03bc are 4 and \u2013 1 respectively<\/p>\n\n\n\n

39. Find the value of x for which the matrix product<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

We know,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

is identity matrix of size 3.<\/p>\n\n\n\n

So according to the given criteria<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we will multiply the two matrices on LHS using the formula cij<\/sub> = ai1<\/sub>b1j<\/sub> + ai2<\/sub>b2j<\/sub> + \u2026 + ain <\/sub>bnj<\/sub>, we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal<\/p>\n\n\n\n

So we get<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

So the value of x is
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\n\n\n\n

Exercise 5.4 Page No: 5.54<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(i) (2A)T<\/sup> = 2 AT<\/sup><\/strong><\/p>\n\n\n\n

(ii) (A + B)T<\/sup> = AT<\/sup> + BT<\/sup><\/strong><\/p>\n\n\n\n

(iii) (A \u2013 B)T<\/sup> = AT<\/sup> \u2013 BT<\/sup><\/strong><\/p>\n\n\n\n

(iv) (AB)T<\/sup> = BT<\/sup> AT<\/sup><\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Put the value of A<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

(ii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n<\/div><\/div>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

Hence proved.<\/p>\n\n\n\n

(iii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

(iv) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

So,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

(i) A + B)T<\/sup> = AT<\/sup> + BT<\/sup><\/strong><\/p>\n\n\n\n

(ii) (AB)T<\/sup> = BT<\/sup> AT<\/sup><\/strong><\/p>\n\n\n\n

(iii) (2A)T<\/sup> = 2 AT<\/sup><\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

(i) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

So,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(ii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

So,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(iii) Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

L.H.S = R.H.S<\/p>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Now we have to find (AB)T<\/sup><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\n\n\n\n

Exercise 5.5 Page No: 5.60<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

 \u2026 (i)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

 \u2026 (ii)<\/p>\n\n\n\n

From (i) and (ii) we can see that<\/p>\n\n\n\n

A skew-symmetric matrix is a square matrix whose transpose equal to its negative, that is,<\/p>\n\n\n\n

X = \u2013 XT<\/sup><\/p>\n\n\n\n

So, A \u2013 AT<\/sup> is a skew-symmetric.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Consider,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 \u2026 (i)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

 \u2026 (ii)<\/p>\n\n\n\n

From (i) and (ii) we can see that<\/p>\n\n\n\n

A skew-symmetric matrix is a square matrix whose transpose equals its negative, that is,<\/p>\n\n\n\n

X = \u2013 XT<\/sup><\/p>\n\n\n\n

So, A \u2013 AT<\/sup> is a skew-symmetric matrix.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 is a symmetric matrix.<\/p>\n\n\n\n

We know that A = [aij<\/sub>]m \u00d7 n <\/sub>is a symmetric matrix if aij <\/sub>= aji<\/sub><\/p>\n\n\n\n

So,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Hence, x = 4, y = 2, t = -3 and z can have any value.<\/p>\n\n\n\n

4. Let. Find matrices X and Y such that X + Y = A, where X is a symmetric and y is a skew-symmetric matrix.<\/strong><\/p>\n\n\n\n

Solution:<\/strong><\/p>\n\n\n\n

Given,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 Then
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Now,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

X is a symmetric matrix.<\/p>\n\n\n\n

Now,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

-Y T<\/sup> = Y<\/p>\n\n\n\n

Y is a skew symmetric matrix.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Hence, X + Y = A<\/p>\n\n\n\n

RD Sharma Solutions for Class 12 Maths Chapter 5: Download PDF<\/strong><\/h2>\n\n\n\n

RD Sharma Solutions for Class 12 Maths Chapter 5\u2013Algebra of Matrices<\/p>\n\n\n\n

Download PDF<\/strong>: RD Sharma Solutions for Class 12 Maths Chapter 5\u2013Algebra of Matrices PDF<\/a><\/p>\n\n\n\n

Chapterwise RD Sharma Solutions for Class 12 Maths :<\/strong><\/h2>\n\n\n\n