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

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

Given $A=\left[\begin{array}{ll}1& 3\\ 2& 2\end{array}\right],I=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]$, if $A-\lambda I$ is a singular matrix then$.......$

Consider the matrices $A=\left[\begin{array}{ccc}4& 6& -1\\ 3& 0& 2\\ 1& -2& 5\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 4\\ 0& 1\\ -1& 2\end{array}\right],C=\left[\begin{array}{l}3\\ 1\\ 2\end{array}\right]$ out of the given matrix product$.....$
i) $(AB{)}^{T}C$
ii) $C^{T}C{\left(AB\right)}^T$
iii) $C^{T}AB$
iv) $A^{T}AB{B}^{T}C$.

If $A$ and $B$ are square matrices of equal order, then which one is correct among the following?

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying the equation $A{A}^T=9I$ where $I$ is the identity matrix of order $3,$ then the ordered pair $(a, b)$ is equal to