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.

Show that the matrix \(A=\begin{bmatrix}1 & -1 & 5 \\ -1 & 2 & 1\\5 & 1 & 3
Show that the matrix \(A=\begin{bmatrix}1 & -1 & 5 \\ -1 & 2 & 1\\5 & 1 & 3

Class 12

Matrices

Exercise 3.3

1. Find the transpose of each of the following matrices:

(i) \(\begin{bmatrix}5 \\ {1\over 2}\\-1 \end{bmatrix}\)

(ii) \(\begin{bmatrix}1 &-1 \\ 2 & 3 \end{bmatrix}\)

(iii) \(\begin{bmatrix}-1 & 5 & 6 \\ {\sqrt 3} & 5 & 6 \\2 & 3 & -1\end{bmatrix}\)

2. If  \(A=\begin{bmatrix}-1 & 2 & 3 \\ 5 & 7 & 9 \\-2 & 1 & 1\end{bmatrix}\) and \(B=\begin{bmatrix}-4 & 1 & -5 \\ 1 & 2 & 0 \\1 & 3 & 1\end{bmatrix}\), then verify that 

(i) (A+B)’=A’+B’

(ii) (A-B)’=A’-B’

3. If \(A’=\begin{bmatrix}3 & 4 \\ -1 & 2 \\0 & 1\end{bmatrix}\) and \(B=\begin{bmatrix}-1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}\), then verify that

(i) (A+B)’=A’+B’

(ii) (A-B)’=A’-B’

4. If \(A’=\begin{bmatrix}-2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B=\begin{bmatrix}-1 & 0 \\ 1 & 2 \end{bmatrix}\), then find (A+2B)’.

5. For the matrices A and B, verify that (AB)′ = B′A′, where 

(i) \(A=\begin{bmatrix}1 \\ -4 \\ 3 \end{bmatrix}\), \(B=\begin{bmatrix}-1 & 2 & 1\end{bmatrix}\)

(ii) \(A=\begin{bmatrix}0 \\ 1 \\ 2 \end{bmatrix}\), \(B=\begin{bmatrix}1 & 5 & 7\end{bmatrix}\) 

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.

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.

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.

9. Find  \({1\over 2}(A+A’)\) and \({1\over 2}(A-A’)\), when \(A=\begin{bmatrix}0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}\).

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:

(A) \(\pi \over 6\)

(B) \(\pi \over 3\)

(C) \(\pi\)

(D) \({3\pi \over 2}\)