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:

If $A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & -4 \\ -3 & r & s \end{bmatrix}$ is a skew-symmetric matrix,then $|A| + |B| - |AB| = $

Let $A_1, A_2, A_3$ be three $A$.$P$.s with the same common difference $d$ and having their first terms as $A, A+1, A+2$,respectively. Let $a, b, c$ be the $7^{\text{th}}, 9^{\text{th}}, 17^{\text{th}}$ terms of $A_1, A_2, A_3$,respectively,such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$. If $a=29$,then the sum of the first $20$ terms of an $A$.$P$. whose first term is $c-a-b$ and common difference is $\frac{d}{12}$,is equal to $........$.

Matrix $A$ satisfies $A^2 = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,$A^n$ is equal to $(n \in N)$:

Let $X = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$. Let $Y$ be a $2 \times 2$ real matrix satisfying the condition $XY = YX$. Then the smallest possible value of $\det(Y)$ is

If both $\left( A - \frac{I}{2} \right)$ and $\left( A + \frac{I}{2} \right)$ are orthogonal matrices,then: