Matrices Questions

If $A$ and $B$ are square matrices of size $n\times n$ such that $A^2-B^2=(A-B)(A+B)$, then which of the following will be always true?

Let $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$ and $B=\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right)$, $a,b\in \mathbb{N}$. Then:

If $D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$ for $x\ne 0,y\ne 0$, then $D$ is:

Let $A=\left[\begin{array}{cc}5\alpha & \alpha \\ 0& \alpha \end{array}\right]$. If $|A^2|=25$, then $|\alpha |$ equals:

Let $P$ and $Q$ be $3\times 3$ matrices with $P\ne Q$. If $P^3=Q^3$ and $P^{2}Q=Q^{2}P$, then determinant of $(P^2+Q^2)$ is equal to:

If $A^2-A+I=0$, then the inverse of $A$ is:

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]$. If $u_1$ and $u_2$ are column matrices such that $A{u}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ and $A{u}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, then $u_1+u_2$ is equal to:

Let $A$ and $B$ be two symmetric matrices of order $3$.Statement-1 : $ABA$ and $A{B}^{2}A$ are symmetric matrices.Statement-2 : $AB$ is a symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$, and $2x+2y+z=0$ possess a non-zero solution is:

Consider the system of linear equations:$x_1+2x_2+x_3=3$$2x_1+3x_2+x_3=3$$3x_1+5x_2+2x_3=1$ The system has:

Let $A$ be a $2\times 2$ matrix with non-zero entries and let $A^2=I$.Statement-1: $\text{Tr}(A)=0$ Statement-2: $|A|=1$

The number of $3\times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$, is:

1: $\text{adj}(\text{adj }A)=A$ where $A$ is a $2\times 2$ matrix.
2: $|\text{adj }A|=|A|$.

Let $A$ be a square matrix all of whose entries are integers. Then which one of the following is true?

Let $A$ be a $2\times 2$ matrix with real entries. Let $I$ be the $2\times 2$ identity matrix. Denote by $\text{tr}(A)$ the sum of diagonal entries of $A$. Assume $A^2=I$.Statement - 1: If $A\ne I$ and $A\ne -I$, then $\det A=-1$.Statement - 2: If $A\ne I$ and $A\ne -I$, then $\text{tr}(A)\ne 0$.

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then$\Delta =\left|\begin{array}{ccc}1& {\omega }^n& {\omega }^{2n}\\ {\omega }^n& {\omega }^{2n}& 1\\ {\omega }^{2n}& 1& {\omega }^n\end{array}\right|$ is equal to:

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then:

If $a_1,a_2,a_3,\dots ,a_n,\dots$ are in G.P., then the determinant $\Delta =\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$ is equal to:

If $a^2+b^2+c^2=-2$ and $f(x)=\left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& 1+c^{2}x\end{array}\right|$, then $f(x)$ is a polynomial of degree:

If $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then which one of the following holds for all $n\ge 1$?

The system of equations:$\alpha x+y+z=\alpha -1$,$x+\alpha y+z=\alpha -1$,$x+y+\alpha z=\alpha -1$ has no solution, if $\alpha$ is:

Let $A=\left(\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right).$ The only correct statement about the matrix $A$ is

Let $A=\left(\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right)$ and $10B=\left(\begin{array}{ccc}4& 2& 2\\ -5& 0& a\\ 1& -2& 3\end{array}\right).$ If $B$ is the inverse of matrix $A,$ then $a$ is