Write A^ -1 for A= 2 & 5 1 & 3 - Vidyadip

Question

Write $A^{-1}$ for $\text{A}=\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$

✓

Answer

$|\text{A}|=\begin{bmatrix} 2 \ \&\ 5 \\ 1 \& 3 \end{bmatrix}=1\neq0$ Let $C_{ij}$ be the cofactor of $a_{ij}$ in $A.$
The cofactors of element $A$ are given by
$C_{11} = 3$
$C_{12} = -1$
$C_{21} = -5$
$C_{22} = 2$
$\text{adj A}=\begin{bmatrix} 3\ \&\ -1 \\ -5\ \&\ 2 \end{bmatrix}^\text{T}=\begin{bmatrix} 3\ \&\ -5 \\ -1 & 2 \end{bmatrix}$
$|\text{A}|=6-5=1$
$\therefore\text{A}^{-1}=\frac{1}{|\text{A}|}\text{ adj A}=\begin{bmatrix} 3\ \&\ -5 \\ -1\ \&\ 2 \end{bmatrix}$

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