NCERT Solutions For Class 12 Maths Chapter 3 Matrices Ex 3.3
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Textbook | NCERT |
Board | CBSE |
Category | NCERT Solutions |
Class | Class 12 |
Subject | Maths |
Chapter | Chapter 3 |
Exercise | Class 12 Maths Chapter 3 Matrices Exercise 3.3 |
Number of Questions Solved | 12 |

NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3
NCERT TEXTBOOK EXERCISES
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 $\left[ \begin{matrix} 5 \ \frac { 1 }{ 2 } \ -1 \end{matrix} \right]$
∴ transpose of A = A’ = [5 1/2−1]

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:
$A=\left[ \begin{matrix} -1 & 2 & 3 \ 5 & 7 & 9 \ -2 & 1 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} -4 & 1 & -5 \ 1 & 2 & 0 \ 1 & 3 & 1 \end{matrix} \right]$


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:
$A’=\left[ \begin{matrix} 3 & 4 \ -1 & 2 \ 0 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} -1 & 2 & 1 \ 1 & 2 & 3 \end{matrix} \right]$


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:
$A’=\begin{bmatrix} -2 & 3 \ 1 & 2 \end{bmatrix},B=\begin{bmatrix} -1 & 0 \ 1 & 2 \end{bmatrix}$

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′=[1 −4 3]



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) $A=\begin{bmatrix} sin\alpha & \quad cos\alpha \ -sin\alpha & \quad cos\alpha \end{bmatrix}$
$A’=\begin{bmatrix} cos\alpha & \quad -sin\alpha \ sin\alpha & \quad cos\alpha \end{bmatrix}$

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 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, $A=\begin{bmatrix} 1 & 5 \ 6 & 7 \end{bmatrix}$
(i) (A+A’) is a symmetric matrix.
(ii) (A-A’) is a skew-symmetric matrix.
Solution:
$A=\begin{bmatrix} 1 & 5 \ 6 & 7 \end{bmatrix}$
=> $A’=\begin{bmatrix} 1 & 6 \ 5 & 7 \end{bmatrix}$

Ex 3.3 Class 12 Maths Question 9.
Find 1/2(A+A′) and 1/2(A−A′),when
$A=\left[ \begin{matrix} 0 & a & b \ -a & 0 & c \ -b & -c & 0 \end{matrix} \right]$
Solution:
$A=\left[ \begin{matrix} 0 & a & b \ -a & 0 & c \ -b & -c & 0 \end{matrix} \right]$
$A’=\left[ \begin{matrix} 0 & -a & -b \ a & 0 & -c \ b & c & 0 \end{matrix} \right]$

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 $\begin{bmatrix} 3 & 5 \ 1 & -1 \end{bmatrix}$
=> $\begin{bmatrix} 3 & 1 \ 5 & -1 \end{bmatrix}$






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 $A=\begin{bmatrix} cos\alpha & \quad -sin\alpha \ sin\alpha & \quad cos\alpha \end{bmatrix}$ then A+A’ = I, if the
value of α is
(a) π/6
(b) π/3
(c) π
(d) 3π/2
Solution:
Now

Thus option (b) is correct.
NCERT Solutions for Class 12 Maths Chapter 3 Matrices Exercise 3.3 PDF
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