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Matrics — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMatrics4 Marks
Question
Find the inverse of the following matrices by the adjoint method : $\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right]$
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Answer

Let $ A=\left[\begin{array}{cc}2 & -2 \\ 4 & 3\end{array}\right] \\ |A|==6+8=14 \neq 0 $
$ \therefore A^{-1} $ exist  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} A=\left[\begin{array}{cc}3 & 2 \\ -4 & 2\end{array}\right]$
$\therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{14}\left(\begin{array}{cc}3 & 2 \\ -4 & 2\end{array}\right)$

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