Express the following matrix as the sum of a symmetric and a skew symmetric matrix:$\left[\begin{array}{ccc}3& 3& -1\\ -2& -2& 1\\ -4& -5& 2\end{array}\right]$,

Let $A=\left[a_{ij}\right]$ is a square matrix of order $2$ where $a_{ij}=i^2-j^2$. Then $A$ is

If $A=\left[\begin{array}{rrr}-1& 1& 2\\ 3& 0& 4\\ 2& 1& -2\end{array}\right]$ and $B=\left[\begin{array}{rrr}2& -3& -4\\ -1& 0& -1\\ 0& -1& 0\end{array}\right]$, then examine whether the matrix $A^2-2B$ is singular.

Using elementary transformation find the inverse of the following matrix:

$A=\left[\begin{array}{rrr}1& 3& -2\\ -3& 0& -5\\ 2& 5& 0\end{array}\right]$