Define a symmetric matrix. Prove that for $\mathrm{A}=\left[\begin{array}{cc}2& 4\\ 5& 6\end{array}\right]$, $\mathrm{A}+\mathrm{A}\text{'}$ is a symmetric.

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

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

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

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

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

$\left[\begin{array}{ccc}3& 2& 5\\ 4& 1& 3\\ 0& 6& 7\end{array}\right]$

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

$\left[\begin{array}{ccc}2& 4& -6\\ 7& 3& 5\\ 1& -2& 4\end{array}\right]$

Use elementary column operation ${\mathrm{C}}_2\to {\mathrm{C}}_2+2{\mathrm{C}}_1$ in the following matrix equation :

$\left[\begin{array}{cc}2& 1\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}3& 1\\ 2& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ -1& 0\end{array}\right]$

Find the inverse of the following matrices, if they exist, by using elementary operations.

$\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$