Find the transpose of each of the following matrices:

(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) $\left(\begin{array}{ccc}-1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1\end{array}\right)$

If $A=\left(\begin{array}{ccc}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right)$, then verify that

(i) $(A+B)^{\prime}=A^{\prime}+B^{\prime}$

(ii) $(A-B)^{\prime}=A^{\prime}-B^{\prime}$

If $A^{\prime}=\left(\begin{array}{cc}3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right)$ and $B=\left(\begin{array}{ccc}-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right)$, then verify that

(i) $(A+B)^{\prime}=A^{\prime}+B^{\prime}$

(ii) $(A-B)^{\prime}=A^{\prime}-B^{\prime}$

If $A^{\prime}=\left(\begin{array}{cc}-2 & 3 \\ 1 & 2\end{array}\right)$ and $B=\left(\begin{array}{cc}-1 & 0 \\ 1 & 2\end{array}\right)$, then find $(A+2 B)^{\prime}$.

For the matrices $A$ and $B$, verify that $(A B)^{\prime}=B^{\prime} A^{\prime}$ where

(i)

(ii)

If

(i) $\quad A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$, then verify $A^{\prime} A=I$

(ii) $A=\left(\begin{array}{cc}\sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha\end{array}\right)$, then verify $A^{\prime} A=I$

(i) Show that the matrix $A=\left(\begin{array}{ccc}1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3\end{array}\right)$ is a symmetric matrix.

(ii) Show that the matrix $A=\left(\begin{array}{ccc}0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0\end{array}\right)$ is a skew symmetric matrix.

For the matrix $A=\left(\begin{array}{ll}1 & 5 \\ 6 & 7\end{array}\right)$, verify that

(i) $\left(A+A^{\prime}\right)$ is a symmetric matrix.

(ii) $\left(A-A^{\prime}\right)$ is a skew symmetric matrix.

Find $\frac{1}{2}\left(A+A^{\prime}\right)$ and $\frac{1}{2}\left(A-A^{\prime}\right)$, when $A=\left(\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right)$.

Solution:

It is given that

Hence,

Now,

Therefore,

Now,

Thus,

Express the following as the sum of a symmetric and skew symmetric matrix:

(i) $\quad\left(\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right)$

(ii) $\quad\left(\begin{array}{ccc}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right)$

(iii) $\left(\begin{array}{ccc}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right)$

(iv) $\quad\left(\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right)$

Solution:

(i) Let $A=\left(\begin{array}{cc}3 & 5 \\ 1 & -1\end{array}\right)$

Hence,

Now,

Let

Now,

Thus, $P=\frac{1}{2}\left(A+A^{\prime}\right)$ is a symmetric matrix.

Now,

Let

Now,

Thus, $Q=\frac{1}{2}\left(A-A^{\prime}\right)$ is a skew symmetric matrix.

Representing $A$ as the sum of $P$ and $Q$ :

(ii) Let

Hence,

Now,

Let

Now,

Thus, $P=\frac{1}{2}\left(A+A^{\prime}\right)$ is a symmetric matrix.

Now,

Let

Now,

Thus, $Q=\frac{1}{2}\left(A-A^{\prime}\right)$ is a skew symmetric matrix.

Representing $A$ as the sum of $P$ and $Q$ :

(iii) Let

Hence,

Now,

Let

Now,

Thus, $P=\frac{1}{2}\left(A+A^{\prime}\right)$ is a symmetric matrix.

Now,

Let

Now,

Thus, $Q=\frac{1}{2}\left(A-A^{\prime}\right)$ is a skew symmetric matrix.

Representing $A$ as the sum of $P$ and $Q$ :

(iv) Let $A=\left(\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right)$

Hence,

Now,

Let

Now,

Thus, $P=\frac{1}{2}\left(A+A^{\prime}\right)$ is a symmetric matrix.

Now,

Let

Now,

Thus, $Q=\frac{1}{2}\left(A-A^{\prime}\right)$ is a skew symmetric matrix.

Representing $A$ as the sum of $P$ and $Q$ :

If $A, B$ are symmetric matrices of the same order, then $A B-B A$ is a

(A) Skew symmetric matrix

(B) Symmetric matrix

(C) Zero matrix

(D) Identity matrix

If $A=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right)$, then $A+A^{\prime}=I$, if the value of $\alpha$ is:

(A) $\frac{\pi}{6}$

(B) $\frac{\pi}{3}$

(C) $\pi$

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