Question
Find the inverse of the following matrices by the adjoint method : $\left[\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right]$
✓
Answer
Let $A=\left[\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right]$$\therefore|A|=\left|\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right|=-2+15=13 \neq 0$
$\therefore A^{-1}$ exists.
First we have to find the co-factor matrix
$= [A_{ij}]_{2\times 2}$, where $A_{ij} = (-1)^{i+j}M_{ij}$
$Now, A_{11} = (-1)^{1+1}M_{11} = 2$
$A_{12} = (-1)^{1+2}M_{12} = -(-3) = 3$
$A_{21} = (-1)^{2+1}M_{21} = -5$
$A_{22} = (-1)^{2+2}M_{22} = -1$
Hence, the co-factor matrix
$=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]=\left[\begin{array}{cc}2 & 3 \\ -5 & -1\end{array}\right]$
$\therefore$ adj $A=\left[\begin{array}{ll}2 & -5 \\ 3 & -1\end{array}\right]$
$\therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{13}\left[\begin{array}{ll}2 & -5 \\ 3 & -1\end{array}\right]$
$\therefore A^{-1}$ exists.
First we have to find the co-factor matrix
$= [A_{ij}]_{2\times 2}$, where $A_{ij} = (-1)^{i+j}M_{ij}$
$Now, A_{11} = (-1)^{1+1}M_{11} = 2$
$A_{12} = (-1)^{1+2}M_{12} = -(-3) = 3$
$A_{21} = (-1)^{2+1}M_{21} = -5$
$A_{22} = (-1)^{2+2}M_{22} = -1$
Hence, the co-factor matrix
$=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]=\left[\begin{array}{cc}2 & 3 \\ -5 & -1\end{array}\right]$
$\therefore$ adj $A=\left[\begin{array}{ll}2 & -5 \\ 3 & -1\end{array}\right]$
$\therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{13}\left[\begin{array}{ll}2 & -5 \\ 3 & -1\end{array}\right]$
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