If $T = \begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}$ , find $T^{-1}$ , the inverse of T.

$\begin{pmatrix} -8 & -5 \\ 3 & 2 \end{pmatrix}$

$\begin{pmatrix} -8 & -5 \\ 3 & -2 \end{pmatrix}$

$\begin{pmatrix} -8 & -5 \\ -3 & 2 \end{pmatrix}$

$\begin{pmatrix} -8 & -5 \\ -3 & -2 \end{pmatrix}$

Correct answer is A

Let $\begin{pmatrix} a & b \\ c & d \end{pmatrix} = T^{-1}$

$T . T^{-1} = I$

$\begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

$\implies -2a - 5c = 1$

$-2b - 5d = 0 \implies b = \frac{-5d}{2}$

$3a + 8c = 0 \implies a = \frac{-8c}{3}$

$3b + 8d = 1$

$-2(\frac{-8c}{3}) - 5c = \frac{16c}{3} - 5c = \frac{c}{3} = 1 \implies c = 3$

$3(\frac{-5d}{2}) + 8d = \frac{-15d}{2} + 8d = \frac{d}{2} = 1 \implies d = 2$

$b = \frac{-5 \times 2}{2} = -5$

$a = \frac{-8 \times 3}{3} = -8$

$\therefore T^{-1} = \begin{pmatrix} -8 & -5 \\ 3 & 2 \end{pmatrix}$

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If T=(−2−538)T = \begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}, find T−1T^{-1}, the inverse of T.