Matrices and Determinants

Let $M$ and $N$ be two $3\times3$ non-singular skew-symmetric matrices such that $MN=NM$. If $P^T$ denotes the transpose of $P$, then $M^2N^2\left(M^TN\right)^{-1}\left(MN^{-1}\right)^T$ is equal to

If the point $P(a,b,c)$, with reference to (E), lies on the plane $2x+y+z=1$, then the value of $7a+b+c$ is

Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying (E), then the value of
$$\frac3{\omega^a}+\frac1{\omega^b}+\frac3{\omega^c}$$
is equal to

Let $\omega\ne1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form
$$\begin{bmatrix}1&a&b\\\omega&1&c\\\omega^2&\omega&1\end{bmatrix},$$
where each of $a$, $b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

Let $M$ be a $3\times3$ matrix satisfying
$$M\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}-1\\2\\3\end{bmatrix},\quad M\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}1\\1\\-1\end{bmatrix},\quad\text{and}\quad M\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}0\\0\\12\end{bmatrix}.$$
Then the sum of the diagonal entries of $M$ is

Match the statements given in Column I with the intervals/union of intervals given in Column II

The number of $3\times3$ matrices A whose entries are either 0 or 1 and for which the system $A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$ has exactly two distinct solutions, is

The number of A in $T_p$ such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is

The number of A in $T_p$ such that the trace of A is not divisible by p but det (A) is divisible by p is

[Note: The trace of a matrix is the sum of its diagonal entries.]

The number of A in $T_p$ such that det (A) is not divisible by p is

Let $\omega$ be the complex number $\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3$. Then the number of distinct complex numbers z satisfying $\begin{vmatrix}z+1&\omega&\omega^2\\\omega&z+\omega^2&1\\\omega^2&1&z+\omega\end{vmatrix}=0$ is equal to

The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations
$$(y+z)\cos3\theta=(xyz)\sin3\theta$$
$$x\sin3\theta=\frac{2\cos3\theta}y+\frac{2\sin3\theta}z$$
$$(xyz)\sin3\theta=(y+2z)\cos3\theta+y\sin3\theta$$
have a solution $(x_0,y_0,z_0)$ with $y_0z_0\ne0$, is

Let $k$ be a positive real number and let $$A=\begin{bmatrix}2k-1&2\sqrt k&2\sqrt k\\2\sqrt k&1&-2k\\-2\sqrt k&2k&-1\end{bmatrix}\ \text{ and }\ B=\begin{bmatrix}0&2k-1&\sqrt k\\1-2k&0&2\sqrt k\\-\sqrt k&-2\sqrt k&0\end{bmatrix}.$$ If $\det(\operatorname{adj}A)+\det(\operatorname{adj}B)=10^6$, then $[k]$ is equal to

[Note: adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$].

The number of matrices in $\mathcal A$ is

The number of matrices $A$ in $\mathcal A$ for which the system of linear equations
$$A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$$
has a unique solution, is

The number of matrices $A$ in $\mathcal A$ for which the system of linear equations
$$A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$$
is inconsistent, is

Match the statements/expressions given in Column I with the values given in Column II.

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$$3x-y-z=0,\qquad-3x+z=0,\qquad-3x+2y+z=0.$$
Then the number of such points for which $x^2+y^2+z^2\le100$ is

Consider the system of equations
$$x-2y+3z=-1,\qquad -x+y-2z=k,\qquad x-3y+4z=1.$$

STATEMENT-1: The system of equations has no solution for $k\ne3$.

and

STATEMENT-2: The determinant $\begin{vmatrix}1&3&-1\\-1&-2&k\\1&4&1\end{vmatrix}\ne0$, for $k\ne3$.

Consider the system of equations $ax+by=0$, $cx+dy=0$, where $a,b,c,d\in\{0,1\}$.

STATEMENT-1: The probability that the system of equations has a unique solution is $\dfrac38$.

and

STATEMENT-2: The probability that the system of equations has a solution is 1.