Properties of Matrices and Inverse

If $A$ is a matrix of order $3\times 3$, then ${\left(A^2\right)}^{-1}$ is equal to

If $A^{-1}=\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right],B^{-1}=\left[\begin{array}{cc}1& 0\\ -3& 1\end{array}\right]$ then ${\left(AB\right)}^{-1}=$

Consider the following statements:

$1.$The inverse of a square matrix, if it exists, is unique.

$2.$If $A \& B$ are singular matrices of order $n,$ then $AB$ is also a singular matrix of order $n.$

Which of the statements given above is/are correct.

If $A$ and $B$ are two non-singular square matrices obeying commutative rule of multiplication then $A^3{B}^3{\left(B^2{A}^4\right)}^{-1}A=$

If $A=\left[\begin{array}{lll}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$ and inverse of $A$ is $\frac{1}{2}\left[\begin{array}{ccc}1& -1& 1\\ -8& 6& -2\\ x& -3& 1\end{array}\right]$ then $x=\_\_\_\_\_$.

If $A=\left[\begin{array}{ll}1& 5\\ 6& 7\end{array}\right]$ and $B=\left[\begin{array}{ll}1& 0\\ 1& 1\end{array}\right]$,
then which one of the following is incorrect.

Find the inverse of the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$.

Find the inverse of the matrix $\left[\begin{array}{cccc}5& 0& 0& 0\\ 0& 6& 0& 0\\ 0& 0& 7& 0\\ 0& 0& 0& 8\end{array}\right]$.

Which one of the following is not true when $A$ and $B$ are invertible matrices of same order?

If $A$ and $B$ are invertible matrices of the same order, then $(AB{)}^{-1}=$?

 $A$ is an invertible square matrix and $k\ge 0$,  then $(kA{)}^{-1}=$?

If $A$ be a singular matrix, then ${\left(adj\left(A^{-1}\right)\right)}^{-1}$ is equal to

Matrices $A$ and $B$ will be inverse of each other only if

If $A\left(\mathrm{\theta }\right)=\left[\begin{array}{cc}i\sin \theta & \cos \theta \\ \cos \theta & i\sin \theta \end{array}\right]$ is a matrix where $i=\sqrt{-1},$ then which of the following is not true

If $B=\left[\begin{array}{cc}-2& -2\\ -1& 0\end{array}\right]$ and $A$ is a matrix such that $A^{-1}B=B^{-1}$ and $k{A}^{-1}=2B^{-1}+I,$ where $k$ is some scalar and $I$ is the $2\times 2$ identity matrix, then the value of $k$ is

For non-singular square matrices $A, B$ and $C$ of same order, ${\left(A{B}^{-1}C\right)}^{-1}$ is equal to

Let $A$ be any $3\times 3$ invertible matrix. Then which one of the following is not always true?

The inverse of a skew symmetric matrix (if it exists) is -

If the inverse of the matrix $A=\left[\begin{array}{rrr}1& 2& 2\\ 2& -1& 2\\ 2& 2& 1\end{array}\right]$ is $\frac{1}{5}\left[\begin{array}{ccc}-3& 2& 2\\ 2& -3& \alpha \\ 2& 2& -3\end{array}\right]$ then, $\alpha =$