The number of positive integral solutions of the equation $\left| {\begin{array}{*{20}{c}}{{x^3} + 1}&{{x^2}y}&{{x^2}z}\\{x{y^2}}&{{y^3} + 1}&{{y^2}z}\\{x{z^2}}&{y{z^2}}&{{z^3} + 1}\end{array}} \right| = 11$ is

For a $3 \times 3$ matrix $M$,let $\text{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and $\text{trace}(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2A))$,then the value of $|B|+\text{trace}(B)$ equals:

Let $M$ and $N$ be two invertible square matrices over $\mathbb{R}$ of order $2$ such that $N$ is diagonal. Then $M N M^{-1}$ is diagonal . . . . . .

Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = 0$.
Statement-$I$: ${A^{-1}} = \frac{1}{7}(5I - A)$.
Statement-$II$: The polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5(A - 4I)$.

If $\left|\begin{array}{ccc}1 & 2 & 3-\lambda \\ 0 & -1-\lambda & 2 \\ 1-\lambda & 1 & 3\end{array}\right|=A \lambda^3+B \lambda^2+C \lambda+D$, then $D+A=$

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If $A$ is an invertible $3 \times 3$ matrix and $B$ is a $3 \times 4$ matrix,then $A^{-1}B$ is defined.
Statement $-2$ : It is never true that $A + B, A - B$,and $AB$ are all defined.
Statement $-3$ : Every matrix none of whose entries are zero is invertible.
Statement $-4$ : Every invertible matrix is square and has no two rows the same.