Question

Find the inverse of the matrix below. You must do this by hand and show all work to earn full credit. Give exact answers. No graphing calculator!

Answer 1

The given matrix is,

`\:A=\:\begin{pmatrix}3&5\\ \:2&4\end{pmatrix}`

Therefore,

`\begin{gathered} \mathrm{Find\:2x2\:matrix\:inverse\:according\:to\:the\:formula}: \\ \begin{equation*} \quad\begin{pmatrix}a\: & \:b\: \\ c\: & \:d\:\end{pmatrix}^(-1)=\frac{1}{\det\begin{pmatrix}a\: & \:b\: \\ c\: & \:d\:\end{pmatrix}}\begin{pmatrix}d\: & \:-b\: \\ -c\: & \:a\:\end{pmatrix} \end{equation*} \\ =\frac{1}{\det \begin{pmatrix}3&5\\ 2&4\end{pmatrix}}\begin{pmatrix}4&-5\\ -2&3\end{pmatrix} \end{gathered}`

Where,

`\begin{gathered} \det\begin{pmatrix}3 & 5 \\ 2 & 4\end{pmatrix}=(3*4)-(2*5)=12-10=2 \\ \therefore\det\begin{pmatrix}3 & 5 \\ 2 & 4\end{pmatrix}=2 \end{gathered}`

Hence,

`=(1)/(2)\begin{pmatrix}4 & -5 \\ -2 & 3\end{pmatrix}=\begin{pmatrix}(1)/(2)*4 & (1)/(2)*-5 \\ \:(1)/(2)*-2 & (1)/(2)*4\end{pmatrix}=\begin{pmatrix}2 & -(5)/(2) \\ -1 & (3)/(2)\end{pmatrix}`

Therefore, the answer is

`A^(-1)=\begin{pmatrix}2 & -(5)/(2) \\ -1 & (3)/(2)\end{pmatrix}`