Inverse Matrix

If $A^{-1}=\frac{1}{11}\left[\begin{array}{cc}-3& 4\\ 5& -3\end{array}\right]$, then $A=$

If $A=\left[\begin{array}{cc}3& 4\\ 7& 9\end{array}\right]$ and $AB=I$, then $B$ is

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.

Consider the following statements

(i) The inverse of square matrix, if it exist, is unique.

(ii) If $A$ and $B$ are singular matrices of order $n$, then $AB$ is also a singular matrix of order $n$.

Which of the following statements above is/are correct?

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 the inverse of $\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ is $\frac{1}{7}\left[\begin{array}{cc}2& -1\\ x& 3\end{array}\right]$, then $x$ is equal to

If $A$ is an invertible $3\times 3$ matrix such that $A^3+A^2=4\left(A+I\right)$, then which of the following is TRUE?

If the inverse of the matrix $A=\left[\begin{array}{cc}\pi & 22\\ 1& 7\end{array}\right]$ is $k\left[\begin{array}{cc}7& -22\\ -1& \pi \end{array}\right],$ then the value of $k$ is equal to

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 $A=\left[\begin{array}{ll}3& 4\\ 1& 2\end{array}\right]$, then $A^{-1}$ is equal to

If matrix $A=\left[\begin{array}{ll}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, then find $A^{-1}$

If $A=\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$ and $A^{-1}=\left[\begin{array}{ccc}3& -1& 1\\ \alpha & 6& -5\\ \beta & -2& 2\end{array}\right]$, then the values of $\alpha$ and $\beta$ are, respectively

If the product of two non-zero square matrices is a null matrix, then which of the following is true?

Find $D^{-1}$ if $D=\mathrm{diag}[d_1, d_2, d_3,\dots , d_n]$, where $d_i\ne 0 \forall i=1, 2,\dots , n$.

If $A$ is a invertible square matrix and $k$ is a non-negative real number. then $(kA{)}^{-1}=$?

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

If $A=\left(\begin{array}{ccc}1& \lambda & 2\\ 1& 2& 5\\ 2& 1& 1\end{array}\right)$ is not invertible then $\lambda =$?

Let $A$ and $B$ are two non-singular matrices of order $3$ such that $A+B=2I$ and $A^{-1}+B^{-1}=3I$, then $AB$ is equal to (where, $I$ is the identity matrix of order $3$)