$A$ determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that the value of the chosen determinant is positive is:

Matrix $A$ is such that ${A^2} = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,${A^n} = $

Suppose $A$ is a $3 \times 3$ matrix consisting of integer entries that are chosen at random from the set $\{-1000, -999, \ldots, 999, 1000\}$. Let $P$ be the probability that either $A^2 = -I$ or $A$ is diagonal,where $I$ is the $3 \times 3$ identity matrix. Then,

Let $X = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ and $A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}$. For $k \in N$,if $X^{T} A^{k} X = 33$,then $k$ is equal to:

For a $3 \times 3$ matrix $M$,let $\text{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and $\text{trace}(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2A))$,then the value of $|B|+\text{trace}(B)$ equals:

Let the matrices $A$ and $B$ be defined as $A = \begin{bmatrix} 3 & 2 \\ 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 1 \\ 7 & 3 \end{bmatrix}$. Then the value of $\det(2A^9B^{-1})$ is: