Matrices and Determinants

Let A be a $3\times3$ real matrix such that
$$A\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\1\\0\end{pmatrix};\quad A\begin{pmatrix}1\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\0\\1\end{pmatrix}\quad\text{and}\quad A\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}.$$
If $X=(x_1,x_2,x_3)^T$ and I is an identity matrix of order 3, then the system $(A-2I)X=\begin{pmatrix}4\\1\\1\end{pmatrix}$ has :

Let $A=\begin{bmatrix}0&-2\\2&0\end{bmatrix}$. If M and N are two matrices given by $M=\displaystyle\sum_{k=1}^{10}A^{2k}$ and $N=\displaystyle\sum_{k=1}^{10}A^{2k-1}$ then $MN^2$ is :

Let A be a $3\times3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices A having sum of all the entries equal to 5, is ______.

The system of equations
$$-kx+3y-14z=25$$
$$-15x+4y-kz=3$$
$$-4x+y+3z=4$$
is consistent for all $k$ in the set

Let $A=\begin{pmatrix}2&-2\\1&-1\end{pmatrix}$ and $B=\begin{pmatrix}-1&2\\-1&2\end{pmatrix}$. Then the number of elements in the set $\{(n,m):n,m\in\{1,2,\dots,10\}$ and $nA^n+mB^m=I\}$ is ______.

Let A be a $3\times3$ invertible matrix. If $|\operatorname{adj}(24A)|=|\operatorname{adj}(3\operatorname{adj}(2A))|$, then $|A|^2$ is equal to :

The ordered pair $(a,b)$, for which the system of linear equations
$$3x-2y+z=b$$
$$5x-8y+9z=3$$
$$2x+y+az=-1$$
has no solution, is :

If the system of equations
$$\alpha x+y+z=5,\quad x+2y+3z=4,\quad x+3y+5z=\beta$$
has infinitely many solutions, then the ordered pair $(\alpha,\beta)$ is equal to :

Let $X=\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}$, $Y=\alpha I+\beta X+\gamma X^2$ and $Z=\alpha^2I-\alpha\beta X+\left(\beta^2-\alpha\gamma\right)X^2$, $\alpha,\beta,\gamma\in\mathbb R$. If $Y^{-1}=\begin{bmatrix}\frac15&-\frac25&\frac15\\0&\frac15&-\frac25\\0&0&\frac15\end{bmatrix}$, then $(\alpha-\beta+\gamma)^2$ is equal to ______.

Let the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha$, $-\alpha x+y+2z=-\alpha$ be inconsistent. Then $\alpha$ is equal to :

The positive value of the determinant of the matrix A, whose $\operatorname{Adj}(\operatorname{Adj}(A))=\begin{pmatrix}14&28&-14\\-14&14&28\\28&-14&14\end{pmatrix}$, is ______.

Let $f(x)=\begin{vmatrix}a&-1&0\\ax&a&-1\\ax^2&ax&a\end{vmatrix}$, $a\in\mathbb R$. Then the sum of the squares of all the values of $a$, for which $2f'(10)-f'(5)+100=0$, is

Let for some real numbers $\alpha$ and $\beta$, $a=\alpha-i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8\left(\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3\right)x+\bar ay=0$ has more than one solution, then $\dfrac\alpha\beta$ is equal to

Let $A$ and $B$ be two $3\times3$ matrices such that $AB=I$ and $|A|=\dfrac18$. Then $|\operatorname{adj}(B\operatorname{adj}(2A))|$ is equal to

Let A be a matrix of order $2\times2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of A is a prime number $p$, $2<p<8$, then the number of such matrices A is ______.

If the system of linear equations
$$2x+3y-z=-2$$
$$x+y+z=4$$
$$x-y+|\lambda|z=4\lambda-4$$
where $\lambda\in\mathbb R$, has no solution, then

Let $A$ be a matrix of order $3\times3$ and $\det(A)=2$. Then $\det\left(\det(A)\operatorname{adj}\left(5\operatorname{adj}\left(A^3\right)\right)\right)$ is equal to ______.

If the system of linear equations
$$2x-3y=\gamma+5,$$
$$\alpha x+5y=\beta+1,$$
where $\alpha,\beta,\gamma\in\mathbf R$ has infinitely many solutions, then the value of $|9\alpha+3\beta+5\gamma|$ is equal to ______.

Let $A=\begin{pmatrix}1+i & 1\\ -i & 0\end{pmatrix}$ where $i=\sqrt{-1}$. Then, the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n=A\right\}$ is ______.

The probability that a randomly chosen $2\times2$ matrix with all the entries from the set of first 10 primes, is singular, is equal to :