If A= 2 & 3 5 & -2 , write A^ -1 in terms of A. - Vidyadip

Question

If $\text{A}=\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix},$ write $A^{-1}$ in terms of A.

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Answer

$|\text{A}|=\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}=-19\neq0$A is a non-singular matrix. Therefore, it is invertible.
Let $C_{ij}$ be a cofactor of $a_{ij}$ in A.
The cofactors of element A are given by
$C_{11}= -2$
$C_{12} = -5$
$C_{21} = -3$
$C_{22} = 2$
$\text{adj A}=\begin{bmatrix} -2 & -5 \\ -3 & 2 \end{bmatrix}^\text{T}=\begin{bmatrix} -2 & -3 \\ -5 & 2 \end{bmatrix},$
$\therefore\text{A}^{-1}=\frac{1}{|\text{A}|}\text{ adj A}=\begin{bmatrix} \frac{2}{19} & \frac{3}{19} \\ \frac{5}{19} & \frac{-2}{19} \end{bmatrix}$
$\Rightarrow\text{A}^{-1}=\frac{1}{19}\text{A}$

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