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 $A$ and $B$ be two non-singular skew-symmetric matrices such that $AB = BA$. Then $A^{2} B^{2} (A^{\top} B)^{-1} (A B^{-1})^{\top}$ is equal to

Let $A = \begin{bmatrix} 3 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 3 \end{bmatrix}$. Then, the roots of the equation $\operatorname{det}(A - \lambda I_{3}) = 0$ (where $I_{3}$ is the identity matrix of order $3$) are

Let $a, b, c \in \mathbb{R}$ be all non-zero and satisfy $a^{3}+b^{3}+c^{3}=2$. If the matrix $A=\begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$ satisfies $A^{T} A=I$,then a value of $abc$ can be

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

Let $A$ and $B$ be any two $n \times n$ matrices such that the following conditions hold: $A B=B A$ and there exist positive integers $k$ and $l$ such that $A^k=I$ (the identity matrix) and $B^l=0$ (the zero matrix). Then,