Let $[A]_{3 \times 3}$ be a non-singular matrix such that $A^{-1}=\frac{1}{3}(A^2-5A+7I)$. Then $17A^8-85A^7+119A^6-51A^5-19A^4+95A^3-133A^2+58A+I=$

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if

Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ 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 $\operatorname{Tr}(D_1 D_2 A + D_1 D_2 + D_1 A + D_2 A) - \operatorname{Tr}(D_1 + D_2) =$

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = [p_{ij}]$ be an $n \times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $P^2 \neq 0$ when $n =$

Let $P$ be an $m \times m$ matrix such that $P^2=P$. Then,$(I+P)^n$ equals