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If $A=\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right]$, then $2 A-3 A^{-1}=$
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The correct answer is:
$\left[\begin{array}{cc}25 & -15 \\ 25 & -20\end{array}\right]$
$A=\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right] \Rightarrow|A|=-14+15=1$
$\therefore A^{-1}=\left[\begin{array}{cc}-7 & 3 \\ -5 & 2\end{array}\right] \quad$ and $\quad 2 A=\left[\begin{array}{cc}4 & -6 \\ 10 & -14\end{array}\right]$
$\therefore 2 A-3 A^{-1}=\left[\begin{array}{cc}4-(-21) & -6-9 \\ 10-(-15) & -14-6\end{array}\right]=\left[\begin{array}{cc}25 & -15 \\ 25 & -20\end{array}\right]$
$\therefore A^{-1}=\left[\begin{array}{cc}-7 & 3 \\ -5 & 2\end{array}\right] \quad$ and $\quad 2 A=\left[\begin{array}{cc}4 & -6 \\ 10 & -14\end{array}\right]$
$\therefore 2 A-3 A^{-1}=\left[\begin{array}{cc}4-(-21) & -6-9 \\ 10-(-15) & -14-6\end{array}\right]=\left[\begin{array}{cc}25 & -15 \\ 25 & -20\end{array}\right]$
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