If $A=\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right]$, then $A-A^{-1}=$

If $A=\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right]$, then $A-A^{-1}=$
  1. $\left[\begin{array}{cc}-5 & 0 \\ 0 & -5\end{array}\right]$
  2. $\left[\begin{array}{cc}3 & 2 \\ 10 & 3\end{array}\right]$
  3. $3\left[\begin{array}{cc}3 & -2 \\ \frac{10}{3} & -3\end{array}\right]$
  4. $5\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$

Solution

$\begin{aligned} & A-A^{-1}=\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right]-\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right]^{-1}=\left[\begin{array}{ll}2 & -3 \\ 5 & -7\end{array}\right]-\left[\begin{array}{ll}-7 & 3 \\ -5 & 2\end{array}\right] \\ & =\left[\begin{array}{cc}9 & -6 \\ 10 & -9\end{array}\right]=3\left[\begin{array}{cc}3 & -2 \\ \frac{10}{3} & -3\end{array}\right]\end{aligned}$

Asked in: MHT CET 2022 (08 Aug Shift 1)

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