7. (i) Show that the matrix (A=begin{bmatrix}1 & -1 & 5 \ -1 & 2 & 1\5 & 1 & 3 end{bmatrix}) is a symmetric matrix.
(ii) Show that the matrix (A=begin{bmatrix}0 & 1 & -1 \ -1 & 0 & 1\1 & -1 & 0 end{bmatrix}) is a skew symmetric matrix.
Class 12
Matrices
Exercise 3.3
1. Find the transpose of each of the following matrices:
(i) (begin{bmatrix}5 \ {1over 2}\-1 end{bmatrix})
(ii) (begin{bmatrix}1 &-1 \ 2 & 3 end{bmatrix})
(iii) (begin{bmatrix}-1 & 5 & 6 \ {sqrt 3} & 5 & 6 \2 & 3 & -1end{bmatrix})
5. For the matrices A and B, verify that (AB)′ = B′A′, where
(i) (A=begin{bmatrix}1 \ -4 \ 3 end{bmatrix}), (B=begin{bmatrix}-1 & 2 & 1end{bmatrix})
(ii) (A=begin{bmatrix}0 \ 1 \ 2 end{bmatrix}), (B=begin{bmatrix}1 & 5 & 7end{bmatrix})
6. (i) If (A=begin{bmatrix}cosα & sinα \ -sinα & cosα end{bmatrix}), then verify that A’A=I.
(ii) If (A=begin{bmatrix}sinα & cosα \ -cosα & sinα end{bmatrix}), then verify that A’A=I.
8. For the matrix (A=begin{bmatrix}1 & 5 \ 6 & 7 end{bmatrix}), verify that
(i) (A+A’) is a symmetric matrix
(ii) (A-A’) is a skew symmetric matrix.
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) (begin{bmatrix}6 & -2 & 2 \ -2 & 3 & -1 \ 2 & -1 & 3 end{bmatrix})
(iii) (begin{bmatrix}3 & 3 & -1 \ -2 & -2 & 1 \ -4 & -5 & 2 end{bmatrix})
(iv) (begin{bmatrix}1 & 5 \ -1 & 2 end{bmatrix}).
Choose the correct answer in the Exercises 11 and 12.
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
12. If (A=begin{bmatrix}cosα & -sinα \ sinα & cosα end{bmatrix}), then A+A’=I, if the value of α is:
