Students can Download Maths Chapter 3 Matrices Ex 3.3 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

## Karnataka 2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.3

### 2nd PUC Maths Matrices NCERT Text Book Questions and Answers Ex 3.3

Question 1.
(i)
$$\left[\begin{array}{c}{5} \\{\frac{1}{2}} \\{-1}\end{array}\right]$$
(ii)
$$\left[\begin{array}{cc}{1} & {-1} \\{2} & {3}\end{array}\right]$$
(iii)
$$\left[\begin{array}{ccc}{-1} & {5} & {6} \\{\sqrt{3}} & {5} & {6} \\{2} & {3} & {-1}\end{array}\right]$$
Question 2.
$$\mathbf{A}=\left[\begin{array}{ccc}{-1} & {2} & {3} \\{5} & {7} & {9} \\{-2} & {1} & {1}\end{array}\right] \text { and } \mathbf{B}=\left[\begin{array}{ccc}{-4} & {1} & {-5} \\{1} & {2} & {0} \\{1} & {3} & {1}\end{array}\right]$$
then verify that
(i) (A + B)’ = A’ + B’,
(ii) (A – B)’ = A’ – B’

Question 3.
$$\mathbf{A}^{\prime}=\left[\begin{array}{rr}{3} & {4} \\{-1} & {2} \\{0} & {1}\end{array}\right] \text { and } \mathbf{B}=\left[\begin{array}{rrr}{-1} & {2} & {1} \\{1} & {2} & {3}\end{array}\right]$$
(i) (A + B)’ = A’ + B’,
(ii) (A – B)’ = A’ – B’

Question 4.
If $$A^{\prime}=\left[\begin{array}{cc}{-2} & {3} \\{1} & {2}\end{array}\right] \text { and } B=\left[\begin{array}{cc}{-1} & {0} \\{1} & {2}\end{array}\right]$$
then Find (A + 2B)’

Question 5.
For the matrices A and B, verify that (AB)’ = B’A’, where
(i)
$$A=\left[\begin{array}{r}{1} \\{-4} \\{3}\end{array}\right], B=\left[\begin{array}{lll}{-1} & {2} & {1}\end{array}\right]$$
(ii)
$$\mathbf{A}=\left[\begin{array}{l}{\mathbf{0}} \\{\mathbf{1}} {\mathbf{2}}\end{array}\right], \mathbf{B}=\left[\begin{array}{lll}{\mathbf{1}} &{\mathbf{5}} & {\mathbf{7}}\end{array}\right]$$

Question 6.
If (i)
$$\mathbf{A}=\left[\begin{array}{cc}{\cos \alpha} & {\sin \alpha} \\{-\sin \alpha} & {\cos \alpha}\end{array}\right]$$,then verify that A’ A = 1
(ii)
$$\mathbf{A}=\left[\begin{array}{cc}{\sin \alpha} & {\cos \alpha} \\{-\cos \alpha} & {\sin \alpha}\end{array}\right]$$,then verify that A’ A = 1

Question 7.
(i) Show that the matrix $$\mathbf{A}=\left[\begin{array}{rrr}{\mathbf{1}} & {\mathbf{1}} & {\mathbf{5}} \\{-\mathbf{1}} & {\mathbf{2}} & {\mathbf{1}} \\{\mathbf{5}} & {\mathbf{1}} & {\mathbf{3}}\end{array}\right]$$ is a symmentric matrix.
(ii) Show that the matrix
$$\mathbf{A}=\left[\begin{array}{rrr}{0} & {1} & {-1} \\{-1} & {0} & {1} \\{1} & {-1} & {0}\end{array}\right]$$
is a skew symmentric matrix.

Question 8.
For the matrix $$\mathbf{A}=\left[\begin{array}{ll}{\mathbf{1}} & {\mathbf{5}} \\{\mathbf{6}} & {7}\end{array}\right]$$ verify that.
(i) ( A + A’) is a symmetric matrix
(ii)(A – A’) is a skew symmetric matrix

Question 9.
Find $$\frac{1}{2}\left(A+A^{\prime}\right) \text { and } \frac{1}{2}(A-A)$$,when
$$\mathbf{A}=\left[\begin{array}{rrr}{0} & {\mathbf{a}} & {\mathbf{b}} \\{-\mathbf{a}} & {\mathbf{0}} & {\mathbf{c}} \\{-\mathbf{b}} & {-\mathbf{c}} &{\mathbf{0}}\end{array}\right]$$

Question 10.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
(i)
$$\left[\begin{array}{cc}{3} & {5} \\{1} & {-1}\end{array}\right]$$
(ii)
$$\left[\begin{array}{rrr}{6} & {-2} & {2} \\{-2} & {3} & {-1} \\{2} & {-1} & {3}\end{array}\right]$$
(iii)
$$\left[\begin{array}{rrr}{3} & {3} & {-1} \\{-2} & {-2} & {1} \\{-4} & {-5} & {2}\end{array}\right]$$
(iv)
$$\left[\begin{array}{rr}{1} & {5} \\{-1} & {2}\end{array}\right]$$

Choose the correct answer in the Exercise 11 and 12.

Question 11.
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
Ans:
A & B are symmetric matrices A’ = A B’ = B

Question 12.
$$\mathbf{A}=\left[\begin{array}{cc}{\cos \alpha} & {-\sin \alpha} \\{\sin \alpha} & {\cos \alpha}\end{array}\right]$$,then A + A’ = I,, if the value of α is
(A) $$\frac{\pi}{6}$$
(B) $$\frac{\pi}{3}$$
(C) π
(D) $$\frac{3 \pi}{2}$$