The number of singular matrices of order $2 \times 2$,whose elements are from the set $\{2, 3, 6, 9\}$ is

Let $a, b, c$ be non-real numbers satisfying the equation $x^5 = 1$ and $S$ be the set of all non-invertible matrices of the form $\begin{bmatrix} 1 & a & b \\ w & 1 & c \\ w^2 & w & 1 \end{bmatrix}$,where $w = e^{\frac{i 2\pi}{5}}$. Then the number of distinct matrices in the set $S$ is:

Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $MN = NM$. If $P^T$ denotes the transpose of $P$,then $M^2 N^2 (M^T N)^{-1} (M N^{-1})^T$ is equal to

Let $M$ and $N$ be two $3 \times 3$ matrices such that $MN = NM$. Further,if $M \neq N^2$ and $M^2 = N^4$,then:
$(A)$ determinant of $(M^2 + MN^2)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $(M^2 + MN^2)U$ is the zero matrix
$(C)$ determinant of $(M^2 + MN^2) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$,if $(M^2 + MN^2)U$ equals the zero matrix then $U$ is the zero matrix

If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ is expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then which of the following relations is correct?

If $\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \frac{9}{8}(103x+81)$,then $\lambda$ and $\frac{\lambda}{3}$ are the roots of the equation: