# NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3

In this chapter, we provide NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 for English medium students, Which will very helpful for every student in their exams. Students can download the latest NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 pdf, free NCERT solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 book pdf download.

## NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3

Ex 3.3 Class 12 Maths Question 1.
Find the transpose of each of the following matrices:
(i)
(ii)
(iii)
Solution:
(i) let A = $left[ begin{matrix} 5 \ frac { 1 }{ 2 } \ -1 end{matrix} right]$
∴ transpose of A = A’ = $left[ begin{matrix} 5 & frac { 1 }{ 2 } & -1 end{matrix} right]$ Ex 3.3 Class 12 Maths Question 2.
If
then verify that:
(i) (A+B)’=A’+B’
(ii) (A-B)’=A’-B’
Solution: $A=left[ begin{matrix} -1 & 2 & 3 \ 5 & 7 & 9 \ -2 & 1 & 1 end{matrix} right] ,B=left[ begin{matrix} -4 & 1 & -5 \ 1 & 2 & 0 \ 1 & 3 & 1 end{matrix} right]$  Ex 3.3 Class 12 Maths Question 3.
If
then verify that:
(i) (A+B)’ = A’+B’
(ii) (A-B)’ = A’-B’
Solution: $A'=left[ begin{matrix} 3 & 4 \ -1 & 2 \ 0 & 1 end{matrix} right] ,B=left[ begin{matrix} -1 & 2 & 1 \ 1 & 2 & 3 end{matrix} right]$  Ex 3.3 Class 12 Maths Question 4.
If
then find (A+2B)’
Solution: $A'=begin{bmatrix} -2 & 3 \ 1 & 2 end{bmatrix},B=begin{bmatrix} -1 & 0 \ 1 & 2 end{bmatrix}$ Ex 3.3 Class 12 Maths Question 5.
For the matrices A and B, verify that (AB)’ = B’A’, where

Solution: $(i)quad A=left[ begin{matrix} 1 \ -4 \ 3 end{matrix} right]$ $A'=left[ begin{matrix} 1 & -4 & 3 end{matrix} right]$   Ex 3.3 Class 12 Maths Question 6.
If (i) ,the verify that A’A=I
If (ii) ,the verify that A’A=I
Solution:
(i) $A=begin{bmatrix} sinalpha & quad cosalpha \ -sinalpha & quad cosalpha end{bmatrix}$ $A'=begin{bmatrix} cosalpha & quad -sinalpha \ sinalpha & quad cosalpha end{bmatrix}$ Ex 3.3 Class 12 Maths Question 7.
(i) Show that the matrix is a symmetric matrix.
(ii) Show that the matrix is a skew-symmetric matrix.
Solution:
(i) For a symmetric matrix aij = aji
Now, $A=left[ begin{matrix} 1 & -1 & 5 \ -1 & 2 & 1 \ 5 & 1 & 3 end{matrix} right]$ Ex 3.3 Class 12 Maths Question 8.
For the matrix,
(i) (A+A’) is a symmetric matrix.
(ii) (A-A’) is a skew-symmetric matrix.
Solution: $A=begin{bmatrix} 1 & 5 \ 6 & 7 end{bmatrix}$
=> $A'=begin{bmatrix} 1 & 6 \ 5 & 7 end{bmatrix}$  Ex 3.3 Class 12 Maths Question 9.
Find and ,when

Solution: $A=left[ begin{matrix} 0 & a & b \ -a & 0 & c \ -b & -c & 0 end{matrix} right]$ $A'=left[ begin{matrix} 0 & -a & -b \ a & 0 & -c \ b & c & 0 end{matrix} right]$ Ex 3.3 Class 12 Maths Question 10.
Express the following matrices as the sum of a symmetric and a skew-symmetric matrix.
(i)
(ii)
(iii)
(iv)
Solution:
(i) let $A=begin{bmatrix} 3 & 5 \ 1 & -1 end{bmatrix}$
=> $A'=begin{bmatrix} 3 & 1 \ 5 & -1 end{bmatrix}$      Ex 3.3 Class 12 Maths Question 11.
Choose the correct answer in the following questions:
If A, B are symmetric matrices of same order then AB-BA is a
(a) Skew – symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity matrix
Solution:
Now A’ = B, B’ = B
(AB-BA)’ = (AB)’-(BA)’
= B’A’ – A’B’
= BA-AB
= – (AB – BA)
AB – BA is a skew-symmetric matrix Hence, option (a) is correct.

Ex 3.3 Class 12 Maths Question 12.
If then A+A’ = I, if the
value of α is
(a)
(b)
(c) π
(d)
Solution:
Now Thus option (b) is correct.

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