Let $A = \begin{bmatrix} 1+i & 1 \\ -i & 0 \end{bmatrix}$ where $i = \sqrt{-1}$. Then,the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$ is

$A$ and $B$ are $3 \times 3$ matrices such that $AB + A + B = 0$,then:

If $A$,$B$,and $C$ are square matrices of order $3$ such that $A = \begin{bmatrix} x & 0 & 1 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ and $|B| = 36$,$|C| = 4$,$(x, y, z \in \mathbb{N})$ and $|ABC| = 1152$,then the minimum value of $x + y + z$ is

Let $A = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix}$. If $M$ and $N$ are two matrices given by $M = \sum_{k=1}^{10} A^{2k}$ and $N = \sum_{k=1}^{10} A^{2k-1}$,then $MN^2$ 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:

If $\omega$ is a root of the equation $x+\frac{1}{x}+1=0$,then the value of the determinant $\left|\begin{array}{ccc}1 & 1+\omega & 1+\omega+\omega^2 \\ 3 & 4+3 \omega & 5+4 \omega+3 \omega^2 \\ 6 & 9+6 \omega & 11+9 \omega+6 \omega^2\end{array}\right|$ is equal to