find the inverse of the matrix below.\n \begin{bmatrix}16&5\\8&3end{bmatrix}\nif necessary, round to the…

Answer:

$\begin{bmatrix}0.3& - 0.5\-0.8&1.6\end{bmatrix}$

Explanation:

Step1: Recall inverse formula

For a $2\times2$ matrix $\begin{bmatrix}a&b\c&d\end{bmatrix}$, the inverse is $\frac{1}{ad - bc}\begin{bmatrix}d&-b\-c&a\end{bmatrix}$.

Step2: Identify matrix elements

Here $a = 16$, $b = 5$, $c = 8$, $d = 3$.

Step3: Calculate determinant

$ad - bc=16\times3 - 5\times8=48 - 40 = 8$.

Step4: Calculate inverse elements

$\frac{d}{ad - bc}=\frac{3}{8}=0.375\approx0.3$, $\frac{-b}{ad - bc}=\frac{- 5}{8}=-0.625\approx - 0.5$, $\frac{-c}{ad - bc}=\frac{-8}{8}=-1\approx - 0.8$, $\frac{a}{ad - bc}=\frac{16}{8}=2\approx1.6$. So the inverse matrix is $\begin{bmatrix}0.3& - 0.5\-0.8&1.6\end{bmatrix}$.