For matrices $A$ and $B$,if $AB = 4I$,then $A^{-1}$ is equal to:

If $3 A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ and $A A^{T} = I$,then find the value of $\frac{a}{b} + \frac{b}{a}$.

If matrix $A = \frac{1}{11} \begin{bmatrix} -1 & 7 & -24 \\ 2 & a & 4 \\ 2 & -3 & 15 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} 3 & 3 & 4 \\ 2 & -3 & 4 \\ b & -1 & c \end{bmatrix}$,then the values of $a, b, c$ respectively are ......

Show that the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ satisfies the equation $A^{2} - 4A + I = O$,where $I$ is the $2 \times 2$ identity matrix and $O$ is the $2 \times 2$ zero matrix. Using this equation,find $A^{-1}$.

By using elementary operations,find the inverse of the matrix $A=\left[\begin{array}{rr}1 & 2 \\ 2 & -1\end{array}\right]$.

If the multiplicative group consists of $2 \times 2$ matrices of the form $\begin{bmatrix} a & a \\ a & a \end{bmatrix}$,where $a \neq 0$ and $a \in \mathbb{R}$,then the inverse of $\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$ is: