College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 44

Determine the determinant of the matrix $\displaystyle A = \left[ \begin{array}{cc}
2 & 2 \\
1 & -3
\end{array} \right]$ and if possible, the inverse of the matrix.

Using the formula

$\displaystyle |D| = \left[ \begin{array}{cc}
2 & 2 \\
1 & -3
\end{array} \right] = 2 \cdot (-3) - 2 \cdot 1 = -8$


$
\begin{equation}
\begin{aligned}

A^{-1} =& \frac{1}{ad - bc} \left[ \begin{array}{cc}
d & -b \\
-c & a
\end{array} \right]
\\
\\
A^{-1} =& \left[ \begin{array}{cc}
2 & 2 \\
1 & -3
\end{array} \right]^{-1} = \frac{1}{(2)(-3)-(2)(1)} \left[ \begin{array}{cc}
-3 & -2 \\
-1 & 2
\end{array} \right] = \frac{-1}{8} \left[ \begin{array}{cc}
-3 & -2 \\
-1 & 2
\end{array} \right]
= \left[ \begin{array}{cc}
\displaystyle \frac{3}{8} & \displaystyle \frac{1}{4} \\
\displaystyle \frac{1}{8} & \displaystyle \frac{-1}{4}
\end{array} \right]

\end{aligned}
\end{equation}
$