Given A= [ cc 2 & -3 -4 & 7 ], compute A^ -1 and show that 2 A^ -… - Vidyadip

Question

Given $A=\left[\begin{array}{cc}2 & -3 \\ -4 & 7\end{array}\right]$, compute $A^{-1}$ and show that $2 A^{-1}=9 I-A$.

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Answer

$|A|=2,$
$\begin{aligned} A^{-1} & =\frac{1}{2}\left[\begin{array}{ll}7 & 3 \\ 4 & 2\end{array}\right] \\ \text{LHS} & =2 A^{-1}=\left[\begin{array}{ll}7 & 3 \\ 4 & 2\end{array}\right],\end{aligned}$
$\begin{aligned} \text { RHS } & =9\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]-\left[\begin{array}{cc}2 & -3 \\ -4 & 7\end{array}\right] \\ & =\left[\begin{array}{ll}7 & 3 \\ 4 & 2\end{array}\right] \\ \text { LHS } & =\text { RHS }\end{aligned}$

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