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} & {-(2)/(3)} \\ {5} & {-4}\end{bmatrix}\rightarrow\begin{bmatrix}{-(1)/(3)} & {0} \\ {0} & {1}\end{bmatrix} \\ \\ \text{ Eliminate first colum} \\ \\ \begin{bmatrix}{1} & {-(2)/(3)} \\ {0} & {-(2)/(3)}\end{bmatrix}\rightarrow\begin{bmatrix}{-(1)/(3)} & {0} \\ {(5)/(3)} & {1}\end{bmatrix} \\ \\ \begin{bmatrix}{1} & {-(2)/(3)} \\ {0} & {1}\end{bmatrix}\rightarrow\begin{bmatrix}{-(1)/(3)} & {0} \\ {-(5)/(2)} & {-(3)/(2)}\end{bmatrix} \\ \\ \begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix}\rightarrow\begin{bmatrix}{-2} & {-1} \\ {-(5)/(2)} & {-(3)/(2)}\end{bmatrix} \end{gathered}`

So inverse of function is:

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