2nd PUC Maths Question Bank Chapter 3 Matrices Ex 3.3

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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) $$\frac{3 \pi}{2}$$