Question

Find the inverse of the matrix below. Show all work.

Answer 1

Given:

$\begin{bmatrix}{-3} & {2} \\ {5} & {-4}\end{bmatrix}$

Find-: Inverse

Sol:

Unit matrix then:

$\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix}$$\begin{gathered} A\rightarrow I \\ \\ \begin{bmatrix}{-3} & {2} \\ {5} & {-4}\end{bmatrix}\rightarrow\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \\ \\ \end{gathered}$

Convert A as unit matrix then:

$\begin{gathered} A\rightarrow I \\ \\ \begin{bmatrix}{-3} & {2} \\ {5} & {-4}\end{bmatrix}\rightarrow\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \\ \\ \begin{bmatrix}{1} & {-\frac{2}{3}} \\ {5} & {-4}\end{bmatrix}\rightarrow\begin{bmatrix}{-\frac{1}{3}} & {0} \\ {0} & {1}\end{bmatrix} \\ \\ \text{ Eliminate first colum} \\ \\ \begin{bmatrix}{1} & {-\frac{2}{3}} \\ {0} & {-\frac{2}{3}}\end{bmatrix}\rightarrow\begin{bmatrix}{-\frac{1}{3}} & {0} \\ {\frac{5}{3}} & {1}\end{bmatrix} \\ \\ \begin{bmatrix}{1} & {-\frac{2}{3}} \\ {0} & {1}\end{bmatrix}\rightarrow\begin{bmatrix}{-\frac{1}{3}} & {0} \\ {-\frac{5}{2}} & {-\frac{3}{2}}\end{bmatrix} \\ \\ \begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix}\rightarrow\begin{bmatrix}{-2} & {-1} \\ {-\frac{5}{2}} & {-\frac{3}{2}}\end{bmatrix} \end{gathered}$

So inverse of function is:

$=\begin{bmatrix}{-2} & {-1} \\ {-\frac{5}{2}} & {-\frac{3}{2}}\end{bmatrix}$