Question
Find the inverse of the following matrices by the adjoint method : $\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]$
✓
Answer
Let $\mathrm{A}=\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]$
$\begin{array}{l}|A|=6+8=14 \neq 0 \\ \therefore A^{-1} \text { exist }\end{array}$
First we have to find the co $-$ factor matrix
$= [A_{ij}] _{2\times 2}$ where $A_{ij} = (-1)^{i+j}M_{ij}$
Now $A_{11} = (-1)^{1+1}M_{11} = 3$
$A_{12} = (-1)^{1+2}M = -4$
$A_{21} = (-2)^{2+1}M_{21} = (-2) = 2$
$A_{22} = (-1)^{2+2}M_{22} = 2$
Hence the co $-$ factor matrix
$=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]=\left[\begin{array}{cc}3 & -4 \\ 2 & 2\end{array}\right] $
$ \therefore \operatorname{adj} \mathrm{A}=\left[\begin{array}{cc}3 & 2 \\ -4 & 2\end{array}\right] $
$ \therefore \mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|}(\operatorname{adj} \mathrm{A})=\frac{1}{14}\left(\begin{array}{cc}3 & 2 \\ -4 & 2\end{array}\right)$
$\begin{array}{l}|A|=6+8=14 \neq 0 \\ \therefore A^{-1} \text { exist }\end{array}$
First we have to find the co $-$ factor matrix
$= [A_{ij}] _{2\times 2}$ where $A_{ij} = (-1)^{i+j}M_{ij}$
Now $A_{11} = (-1)^{1+1}M_{11} = 3$
$A_{12} = (-1)^{1+2}M = -4$
$A_{21} = (-2)^{2+1}M_{21} = (-2) = 2$
$A_{22} = (-1)^{2+2}M_{22} = 2$
Hence the co $-$ factor matrix
$=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]=\left[\begin{array}{cc}3 & -4 \\ 2 & 2\end{array}\right] $
$ \therefore \operatorname{adj} \mathrm{A}=\left[\begin{array}{cc}3 & 2 \\ -4 & 2\end{array}\right] $
$ \therefore \mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|}(\operatorname{adj} \mathrm{A})=\frac{1}{14}\left(\begin{array}{cc}3 & 2 \\ -4 & 2\end{array}\right)$
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