The matrix $A = \left[ {\begin{array}{*{20}{c}}0&{ - 4}&1\\4&0&{ - 5}\\{ - 1}&5&0\end{array}} \right]$ is:

$A$ and $B$ are two given matrices such that the order of $A$ is $3 \times 4$. If $A'B$ and $BA'$ are both defined,then:

Let $A = \begin{bmatrix} 2 & -2 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 \\ -1 & 2 \end{bmatrix}$. Then the number of elements in the set $\{(n, m) : n, m \in \{1, 2, \ldots, 10\} \text{ and } nA^n + mB^m = I\}$ is

If $A = \begin{bmatrix} \lambda & 1 \\ -1 & -\lambda \end{bmatrix}$,then for what value of $\lambda$ is $A^2 = O$?

The value of $\cos \alpha$ is . . . . . . if $A+A^{\prime}=I$,where $A = \begin{bmatrix} \sin \alpha & -\cos \alpha \\ \cos \alpha & \sin \alpha \end{bmatrix}$.

If $A$ and $B$ are two matrices such that $AB$ is an identity matrix and the order of matrix $B$ is $3 \times 4$,then the order of matrix $A$ is