Find the inverses of the following matrices by the adjoint mathod… - Vidyadip

Question

Find the inverses of the following matrices by the adjoint mathod : $\left[\begin{array}{ll}3 & -1 \\ 2 & -1\end{array}\right]$

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Answer

Let $A =\left(\begin{array}{ll}3 & -1 \\ 2 & -1\end{array}\right)$
Then $|A|=\left|\begin{array}{ll}3 & -1 \\ 2 & -1\end{array}\right|=-3-(-2)=-1 \neq 0$
$\therefore A ^{-1}$ exists.
First we have to find the cofactor matrix $=\left[ A _{i j}\right]_{2 \times 2}$, where $A _{i j}=(-1)^{i+j} M _{i j}$
Now, $A_{11}=(-1)^{1+1} M_{11}=-1$
$
\begin{aligned}
& A_{12}=(-1)^{1+2} M_{12}=-2 \\
& A_{21}=(-1)^{2+1} M_{21}=-(-1)=1 \\
& A_{22}=(-1)^{2+2} M_{22}=3
\end{aligned}
$
$\therefore$ the cofactor matrix
$
\begin{aligned}
& =\left[\begin{array}{ll}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}\right]=\left[\begin{array}{rr}
-1 & -2 \\
1 & 3
\end{array}\right] \\
& \therefore \operatorname{adj} A=\left[\begin{array}{ll}
-1 & 1 \\
-2 & 3
\end{array}\right] \\
& \therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{-1}\left[\begin{array}{ll}
-1 & 1 \\
-2 & 3
\end{array}\right] \\
& \therefore A^{-1}=\left[\begin{array}{ll}
1 & -1 \\
2 & -3
\end{array}\right] .
\end{aligned}
$

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