Let $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$,then:

Let $A$ and $B$ be any two $n \times n$ matrices such that the following conditions hold: $A B=B A$ and there exist positive integers $k$ and $l$ such that $A^k=I$ (the identity matrix) and $B^l=0$ (the zero matrix). Then,

Let $\alpha$ be a root of the equation $(a-c)x^2 + (b-a)x + (c-b) = 0$,where $a, b, c$ are distinct real numbers such that the matrix $\begin{bmatrix} \alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c \end{bmatrix}$ is singular. Then the value of $\frac{(a-c)^2}{(b-a)(c-b)} + \frac{(b-a)^2}{(a-c)(c-b)} + \frac{(c-b)^2}{(a-c)(b-a)}$ is:

If $A$ is a $2 \times 2$ matrix such that $\operatorname{det} A = -21$ and $\operatorname{trace}(A^3) = 2024$, then the trace of $A$ is

If matrix $D_1 = \operatorname{diag}(a, b, c)$,matrix $D_2 = \operatorname{diag}(3, 3, 3)$ and $A$ is a skew-symmetric matrix of $3^{rd}$ order,then $\operatorname{Tr}(D_1 D_2 A + D_1 D_2 + D_1 A + D_2 A) - \operatorname{Tr}(D_1 + D_2) =$

For a $3 \times 3$ matrix $A$,if $A(\operatorname{adj} A) = \begin{bmatrix} -10 & 0 & 0 \\ 0 & -10 & 2 \\ 0 & 0 & -10 \end{bmatrix}$,then the value of the determinant of $A$ is: