EXERCISE 3.3

Find the transpose:

$\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$

Find the transpose:

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

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

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

$\left[\begin{array}{ccc}6& -2& 2\\ -2& 3& -1\\ 2& -1& 3\end{array}\right]$

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

$\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$

Express the following matrices as the sum of a symmetric and a skew-symmetric matrix

$\left[\begin{array}{ll}1& 5\\ -1& 2\end{array}\right]$

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

If $A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$, and $A+A^{\text{'}}=I$, then the value of $\alpha$ is