Let $M$ denote the set of all real matrices of order $3 \times 3$ and let $S=\{-3,-2,-1,1,2\}$. Let $S_1=\{A=[a_{ij}] \in M: A=A^{T} \text{ and } a_{ij} \in S, \forall i, j\}$,$S_2=\{A=[a_{ij}] \in M: A=-A^{T} \text{ and } a_{ij} \in S, \forall i, j\}$,and $S_3=\{A=[a_{ij}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{ij} \in S, \forall i, j\}$. If $n(S_1 \cup S_2 \cup S_3)=125 \alpha$,then $\alpha$ equals.

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 $........$.

Let $A$ be a symmetric matrix and $B$ be a skew-symmetric matrix,such that $A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then $|A|$ is:

For $\alpha, \beta \in R$ and a natural number $n$,let $A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\ 3r - 2 & 3 & \frac{n(3n - 1)}{2} \end{vmatrix}$. Then $2A_{10} - A_8$ is equal to:

If $A = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{-1, 1\} \right\}$,then the number of singular matrices in $A$ is

Let matrix $A = \begin{bmatrix} 5 & -3 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix}$,$X$ be a non-zero matrix of order $3 \times 1$,and $c$ be a real number. If $A^2 X = cAX$,then the number of distinct values of $c$ is: