Question

Find A-¹, if A =
23
32

Answer 1

Given the matrix,
`\begin{bmatrix}2 & 3 \\3 & 2 \end{bmatrix}`, find its inverse.

First we need to find its determinant which can be done by using the following formula,

`det=\begin{bmatrix}a & b \\c & d \end{bmatrix}=ad-cb`

Lets call
`\begin{bmatrix}2 & 3 \\3 & 2 \end{bmatrix}` matrix "A."

So,

`detA=(2)(2)-(3)(3)`

`\Longrightarrow detA=4-9`

`\Longrightarrow detA=-5`

We now have the determinant, we can now use the following formula to determine the inverse matrix,

`A^(-1) =(1)/(detA) \begin{bmatrix}d & -b \\-c & a \end{bmatrix}`

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

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

`\Longrightarrow A^(-1) = \begin{bmatrix}-(2)/(5) & (3)/(5) \\(3)/(5) & -(2)/(5)\end{bmatrix} \therefore \ Sol.`

*Note* I went a few more steps than the question needed. If you back up two steps on my answer is where they stopped. The last option is correct.