Let $A$ be a square matrix such that $AA^T = I$. Then $\frac{1}{2} A[(A+A^T)^2 + (A-A^T)^2]$ is equal to

Let $A = \begin{bmatrix} 4 & -2 \\ \alpha & \beta \end{bmatrix}$. If $A^2 + \gamma A + 18I = O$,then $\operatorname{det}(A)$ is equal to

If $A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$ is expressed as a sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$,then $P^{T}-Q^{T}=$

If $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ is a skew-symmetric matrix and $b, c, f$ are non-zero real numbers, then $\frac{b}{c} = $

The number of ordered pairs $(x, y)$ for which $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 2 & x \\ y & 1 & 2 \end{bmatrix}$ is a singular and symmetric matrix is

Let $A$ denote the matrix $\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$,where $i^2=-1$,and let $I$ denote the identity matrix $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$. Then,$I+A+A^2+\ldots+A^{2010}$ is