Solve the following equations

Question:

If $A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]$, write $A^{-1}$ in terms of $A$

Solution:

$|A|=\left|\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right|=-19 \neq 0$

$A$ is a non-singular matri $x$. Therefore, it is invertible.

Let $C_{i j}$ be a cofactor of $a_{i j}$ in $A$.

The cofactors of element $A$ are given by

$C_{11}=-2$

$C_{12}=-5$

$C_{21}=-3$

$C_{22}=2$

$\operatorname{adj} A=\left[\begin{array}{cc}-2 & -5 \\ -3 & 2\end{array}\right]^{T}=\left[\begin{array}{cc}-2 & -3 \\ -5 & 2\end{array}\right]$

$\therefore A^{-1}=\frac{1}{|A|}$ adj $A=\left[\begin{array}{cc}2 / 19 & 3 / 19 \\ 5 / 19 & -2 / 19\end{array}\right]$

$\Rightarrow A^{-1}=\frac{1}{19} A$

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