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

If $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$, then the absolute value of the determinant of $\left(A^{9}+A^{6}+A^{3}+A\right)$ is __________.

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 _________

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 ___

A $2 \times 2$ matrix is filled with four distinct integers randomly chosen from the set $\{1,2,3,4,5,6\}$. Then the probability that the matrix generated in such a way is singular is

If $A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix}$ is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each other, then the value of determinant of A is:

If $A$ is a $3 \times 3$ non-zero matrix such that $A^2 = 0$ then the determinant of $(I + A)^{50} - 50A$ is equal to