Suppose $a, b$ and $c$ are integers such that $a>b>c>0$, and $A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]$. Then the value of the determinant of $A$ is

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then the determinant of $A + A^2 + A^3 + \cdots + A^{13}$ is

If inverse of the matrix $\left[\begin{array}{cc}2 & -0.5 \\ -1 & x\end{array}\right]$ is $\left[\begin{array}{ll}1 & 1 \\ 2 & 4 \end{array}\right]$, then the value of $x$ is

If $A=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 0\end{array}\right]$ then $A^{2022}$ is

Let $M = \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{bmatrix}$ where $a, b$ and $c$ are real numbers such that $a + b + c = 0$ and $abc \neq 0$. If $\det M = 0$, then the maximum possible value of $\frac{a^2+b^2}{c^2}$ is ___

If $A, B$ and $A + B$ are non singular matrices and $AB = BA$ then $2A - B - A(A + B)^{-1}A + B(A + B)^{-1} B$ equals

If $A = \begin{bmatrix} 2 & n \\ 4 & 1 \end{bmatrix}$ such that $A^3 = 27 \begin{bmatrix} 4 & q \\ p & r \end{bmatrix}$, then $p + q + r$ equals _________