If $A$ is a non-singular matrix of order $3$ such that $(A-2I)(A-4I)=0$,then $\frac{1}{6}A + \frac{4}{3}A^{-1}$ is (where $I$ is a unit matrix of order $3$ and $0$ is a null matrix of order $3$).

How many $3 \times 3$ matrices $M$ with entries from $\{0, 1, 2\}$ are there,for which the sum of the diagonal entries of $M^T M$ is $5$?

If both $\left( A - \frac{I}{2} \right)$ and $\left( A + \frac{I}{2} \right)$ are orthogonal matrices,then:

If $x, y$ are any two non-zero real numbers, $a_{i j} = xi + yj$, $A = \{a_{i j}\}_{n \times n}$ and $P, Q$ are two $n \times n$ matrices such that $A = xP + yQ$, then

$A$ and $B$ are two $3 \times 3$ non-singular matrices such that $\operatorname{adj} A = |A| B$. If $\operatorname{tr}(X)$ denotes the trace of a square matrix $X$ and $C = \begin{bmatrix} 4 & 4 & 7 \\ 3 & -2 & 5 \\ -2 & 3 & 6 \end{bmatrix}$,then $\sum_{k=1}^{\infty} \operatorname{tr}\left(\frac{1}{3^k}(A B)^k C\right)$ is equal to

If a $2^{nd}$ order determinant with elements $0$ or $1$ is chosen from the set of all such determinants,find the probability that the determinant chosen is non-zero.