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 $A = \begin{bmatrix} 3 & -2 \\ 4 & 2 \end{bmatrix}$,then $A^2 - 5A + 14I = 0$. Which of the following is equivalent to $A^2$?

If the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}$ satisfies the equation $A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ for some real numbers $\alpha$ and $\beta$,then $\beta - \alpha$ is equal to ........ .

Consider the following relation $R$ on the set of real square matrices of order $3$. $R = \{(A,B) | A = P^{-1}BP \text{ for some invertible matrix } P\}$.
\textbf{Statement-$1$:} $R$ is an equivalence relation.
\textbf{Statement-$2$:} For any two invertible $3 \times 3$ matrices $M$ and $N$,$(MN)^{-1} = N^{-1}M^{-1}$.

If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$,then which one of the following holds for all $n \ge 1$ (by the principle of mathematical induction)?

For the matrices $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix}$,if $(A^{15}+B)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$,then among the following which one is true?