For $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 1 & 0 \end{bmatrix}$,if $A^3 - 2A^2 + kA - 4I_3 = 0$,then $k = $ . . . . . . .

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ and $M = A + A^{2} + A^{3} + \dots + A^{20}$,then the sum of all the elements of the matrix $M$ is equal to $.....$

Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between:

Suppose the vectors $x_{1}, x_{2}$ and $x_{3}$ are the solutions of the system of linear equations $Ax = b$ when the vector $b$ on the right side is equal to $b_{1}, b_{2}$ and $b_{3}$ respectively. If $x_{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, x_{2} = \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, x_{3} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, b_{1} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, b_{2} = \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}$ and $b_{3} = \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}$,then the determinant of $A$ is equal to

If $A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]$,then $\operatorname{det}\left(A^6+B^6\right)=$

If $a, b, c$ are non-zero complex numbers satisfying $a^2 + b^2 + c^2 = 0$ and $\left| \begin{array}{ccc} b^2 + c^2 & ab & ac \\ ab & c^2 + a^2 & bc \\ ac & bc & a^2 + b^2 \end{array} \right| = k a^2 b^2 c^2$,then $k$ is equal to