find the inverse of the matrix below.\n \begin{bmatrix} 4 & 3 \\ 2 & 6 end{bmatrix} \nif necessary, round to…

Explanation:

Step1: Find the determinant of the matrix

For a (2\times2) matrix (\begin{bmatrix}a&b\c&d\end{bmatrix}), the determinant (D = ad - bc). Here (a = 4), (b = 3), (c = 2), (d = 6). (D=4\times6 - 3\times2=24 - 6=18)

Step2: Use the formula for the inverse of a (2\times2) matrix

The inverse of (\begin{bmatrix}a&b\c&d\end{bmatrix}) is (\frac{1}{D}\begin{bmatrix}d&-b\-c&a\end{bmatrix}) Substitute (a = 4), (b = 3), (c = 2), (d = 6) and (D = 18) (\frac{1}{18}\begin{bmatrix}6&- 3\-2&4\end{bmatrix}=\begin{bmatrix}\frac{6}{18}&\frac{-3}{18}\\frac{-2}{18}&\frac{4}{18}\end{bmatrix}=\begin{bmatrix}\frac{1}{3}&-\frac{1}{6}\-\frac{1}{9}&\frac{2}{9}\end{bmatrix}\approx\begin{bmatrix}0.33&- 0.17\-0.11&0.22\end{bmatrix})

Answer:

The inverse matrix is (\begin{bmatrix}\frac{1}{3}&-\frac{1}{6}\-\frac{1}{9}&\frac{2}{9}\end{bmatrix}) (or approximately (\begin{bmatrix}0.33&-0.17\-0.11&0.22\end{bmatrix}) when rounded to the nearest hundredth)