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

Get Free NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Matrices Exercise 3.3 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 3 Class 12 Matrices Ex 3.3 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

Topics and Sub Topics in Class 11 Maths Chapter 3 Matrices:

Section Name	Topic Name
3	Matrices
3.1	Introduction
3.2	Matrix
3.3	Types of Matrices
3.4	Operations on Matrices
3.5	Transpose of a Matrix
3.6	Symmetric and Skew Symmetric Matrices
3.7	Elementary Operation (Transformation) of a Matrix
3.8	Invertible Matrices

NCERT Solutions for Class 12th Chapter 3 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) $\left[ \begin{matrix} 5 \\ \frac { 1 }{ 2 } \\ -1 \end{matrix} \right]$
(ii) $\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$
(iii) $\left[ \begin{matrix} -1 & 5 & 6 \\ \sqrt { 3 } & 5 & 6 \\ 2 & 3 & -1 \end{matrix} \right]$
Solution:
(i) let A = $\left[ \begin{matrix} 5 \\ \frac { 1 }{ 2 } \\ -1 \end{matrix} \right]$
∴ transpose of A = A’ = $\left[ \begin{matrix} 5 & \frac { 1 }{ 2 } & -1 \end{matrix} \right]$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 1

Ex 3.3 Class 12 Maths Question 2.
If $A=\left[ \begin{matrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix} \right]$
then verify that:
(i) (A+B)’=A’+B’
(ii) (A-B)’=A’-B’
Solution:
$A=\left[ \begin{matrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{matrix} \right]$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 2

Ex 3.3 Class 12 Maths Question 3.
If $A'=\left[ \begin{matrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{matrix} \right]$
then verify that:
(i) (A+B)’ = A’+B’
(ii) (A-B)’ = A’-B’
Solution:
$A'=\left[ \begin{matrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{matrix} \right]$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 3
byjus class 12 maths Chapter 3 Matrices 3.1

Ex 3.3 Class 12 Maths Question 4.
If $A'=\begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix},B=\begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$
then find (A+2B)’
Solution:
$A'=\begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix},B=\begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 4

Ex 3.3 Class 12 Maths Question 5.
For the matrices A and B, verify that (AB)’ = B’A’, where
$(i)\quad A=\left[ \begin{matrix} 1 \\ -4 \\ 3 \end{matrix} \right] ,B=\left[ \begin{matrix} -1 & 2 & 1 \end{matrix} \right]$
$(ii)\quad A=\left[ \begin{matrix} 0 \\ 1 \\ 2 \end{matrix} \right] ,B=\left[ \begin{matrix} 1 & 5 & 7 \end{matrix} \right]$
Solution:
$(i)\quad A=\left[ \begin{matrix} 1 \\ -4 \\ 3 \end{matrix} \right]$
$A'=\left[ \begin{matrix} 1 & -4 & 3 \end{matrix} \right]$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 5

Ex 3.3 Class 12 Maths Question 6.
If (i) $A=\begin{bmatrix} cos\alpha & \quad sin\alpha \\ -sin\alpha & \quad cos\alpha \end{bmatrix}$ ,the verify that A’A=I
If (ii) $A=\begin{bmatrix} sin\alpha & \quad cos\alpha \\ -cos\alpha & \quad sin\alpha \end{bmatrix}$ ,the verify that A’A=I
Solution:
(i) $A=\begin{bmatrix} sin\alpha & \quad cos\alpha \\ -sin\alpha & \quad cos\alpha \end{bmatrix}$
$A'=\begin{bmatrix} cos\alpha & \quad -sin\alpha \\ sin\alpha & \quad cos\alpha \end{bmatrix}$
byjus class 12 maths Chapter 3 Matrices 6

Ex 3.3 Class 12 Maths Question 7.
(i) Show that the matrix $A=\left[ \begin{matrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{matrix} \right]$ is a symmetric matrix.
(ii) Show that the matrix $A=\left[ \begin{matrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{matrix} \right]$ is a skew-symmetric matrix.
Solution:
(i) For a symmetric matrix a_ij = a_ji
Now,
$A=\left[ \begin{matrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{matrix} \right]$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 7

Ex 3.3 Class 12 Maths Question 8.
For the matrix, $A=\begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$
(i) (A+A’) is a symmetric matrix.
(ii) (A-A’) is a skew-symmetric matrix.
Solution:
$A=\begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$
=> $A'=\begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix}$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 8

Ex 3.3 Class 12 Maths Question 9.
Find $\\ \frac { 1 }{ 2 } (A+A')$ and $\\ \frac { 1 }{ 2 } (A-A')$ ,when
$A=\left[ \begin{matrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{matrix} \right]$
Solution:
$A=\left[ \begin{matrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{matrix} \right]$
$A'=\left[ \begin{matrix} 0 & -a & -b \\ a & 0 & -c \\ b & c & 0 \end{matrix} \right]$
byjus class 12 maths Chapter 3 Matrices 9

Ex 3.3 Class 12 Maths Question 10.
Express the following matrices as the sum of a symmetric and a skew-symmetric matrix.
(i) $\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$
(ii) $\left[ \begin{matrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{matrix} \right]$
(iii) $\left[ \begin{matrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{matrix} \right]$
(iv) $\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}$
Solution:
(i) let $A=\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$
=> $A'=\begin{bmatrix} 3 & 1 \\ 5 & -1 \end{bmatrix}$
NCERT Solutions for Class 12 Maths Chapter 3 Matrices 10

byjus class 12 maths Chapter 3 Matrices 10.3

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 $A=\begin{bmatrix} cos\alpha & \quad -sin\alpha \\ sin\alpha & \quad cos\alpha \end{bmatrix}$ then A+A’ = I, if the
value of α is
(a) $\frac { \pi }{ 6 }$
(b) $\frac { \pi }{ 3 }$
(c) π
(d) $\frac { 3\pi }{ 2 }$
Solution:
Now
byjus class 12 maths Chapter 3 Matrices 12
Thus option (b) is correct.