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

Let $A$ be the set of all $3 \times 3$ matrices with entries $0$ or $1$ only. Let $B$ be the subset of $A$ consisting of all matrices with determinant value $1$. Let $C$ be the subset of $A$ consisting of all matrices with determinant value $-1$. Then:

Let $A$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}$ and $A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}$. If $\det(A) = 1$ and the matrix $A$ satisfies $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$,then $\det(\text{adj}(A^2 + A))$ is equal to:

If $A = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$,then the determinant of $A^2 - 2A$ is

If $a, b, c$ are three complex numbers such that $a^2 + b^2 + c^2 = 0$ and $\begin{vmatrix} (b^2 + c^2) & ab & ac \\ ab & (c^2 + a^2) & bc \\ ac & bc & (a^2 + b^2) \end{vmatrix} = K a^2 b^2 c^2$,then the value of $K$ is:

Which of the following statements is correct about two square matrices $A$ and $B$ of the same order $n$?