Given $\left[\begin{array}{cc}4 & 2 \\ -1 & 1\end{array}\right] M =6 I$, where $M$ is a matrix and $I$ is unit matrix of order $2 \times 2$. (a) State the order of matrix M. (b) Find the matrix M.
If $A=\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}-2 & 1 \\ -3 & 2\end{array}\right]$ and $A^2-5 B^2=5 C$, find matrix $C$, where $C$ is a 2 by 2 matrix.
If $A=\left[\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right], B=\left[\begin{array}{cc}4 & -2 \\ -1 & 3\end{array}\right]$ and $I$ is the identity matrix of the same order and $A ^t$ is the transpose of matrix A , find $A ^{ t } B + BI$