If $A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}$,$B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}$ and $(A + B)^2 = A^2 + B^2$,then the values of $a$ and $b$ are:

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then $A^3 - 4A^2 - 6A$ is equal to:

If $A=\left[\begin{array}{ccc}1 & 2 & 3 \\ 1 & 1 & 1 \\ 1 & -1 & 1\end{array}\right], B=\left[\begin{array}{lll}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 4\end{array}\right]$,and $C=\left[\begin{array}{lll}2 & 0 & 1 \\ 0 & 1 & 0 \\ 3 & 2 & 1\end{array}\right]$,then $\left(\left(\left((A B C)^{-1}\right)^T\right)^{-1}\right)^T=$

Let $P$ be a square matrix such that $P^2 = I - P$. For $\alpha, \beta, \gamma, \delta \in N$,if $P^\alpha + P^\beta = \gamma I - 29 P$ and $P^\alpha - P^\beta = \delta I - 13 P$,then $\alpha + \beta + \gamma - \delta$ is equal to $........$.

Let $p, q, r$ be three real numbers satisfying $[p \, q \, r] \begin{bmatrix} 2 & p & q \\ -3 & q & -p+r \\ 12 & r & -q+3r \end{bmatrix} = [5 \, b \, c]$. Then the minimum value of $(b+c)$ is:

$A$ is a $3 \times 3$ matrix satisfying $A^3-5A^2+7A+I=0$. If $A^5-6A^4+12A^3-6A^2+2A+2I=lA+mI$, then $l+m=$