Inverse Matrix

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 $B=\left[\begin{array}{cc}-2& -2\\ -1& 0\end{array}\right]$ 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

The multiplicative inverse of matrix $\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$ is

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

If the product of the matrix $B=\left[\begin{array}{ccc}2& 6& 4\\ 1& 0& 1\\ -1& 1& -1\end{array}\right]\begin{array}{}\\ \\ \end{array}$ with a matrix $A$ has inverse $C=\left[\begin{array}{ccc}-1& 0& 1\\ 1& 1& 3\\ 2& 0& 3\end{array}\right]$, then  $A^{–1}$ equals

The inverse of the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 3& 3& 0\\ 5& 2& -1\end{array}\right]$ is

If $D=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}[d_1,d_2,d_3,\dots , d_n]$, where $d_i\ne 0\forall i=1, 2, \dots , n$ then $D^{-1}$ is equal to

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 inverse of matrix $A=\left[\begin{array}{ccc}-\frac{5}{8}& \frac{5}{4}& \alpha \\ -\frac{3}{8}& \frac{3}{4}& -\alpha \\ -\frac{5}{4}& \frac{3}{2}& 2\alpha \end{array}\right]$ is matrix $B=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& -1\\ 3& -5& 0\end{array}\right]$, then the value of $\sin \frac{2\mathrm{\pi }}{\alpha }$ 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, $\alpha =$

If $A^T{A}^{-1}$ is symmetric, then $A^2=$

If ${\left(AB\right)}^{-1}=\left[\begin{array}{cc}12& -17\\ -19& 27\end{array}\right]$ and $A^{-1}=\left[\begin{array}{cc}1& -1\\ -2& 3\end{array}\right]$ then $B^{-1}=$

If $A$ is a $3\times 3$ non-singular matrix such that $A{A}^T=A^{T}A$ and $B=A^{-1}{A}^T$, then $B{B}^T=$