College Algebra, Chapter 7, 7.3, Section 7.3, Problem 10

Determine the inverse of the matrix $\left[ \begin{array}{cc}
3 & 4 \\
7 & 9
\end{array} \right]$ if it exists.

First, let's add the identity matrix to the right of our matrix

$\left[ \begin{array}{cc}
3 & 4 \\
7 & 9
\end{array} \right]$

By using Gauss-Jordan Elimination

$\displaystyle \frac{1}{3} R_1$

$\left[ \begin{array}{cc|cc}
1 & \displaystyle \frac{4}{3} & \displaystyle \frac{1}{3} & 0 \\
7 & 9 & 0 & 1
\end{array} \right]$

$\displaystyle R_2 - 7 R_1 \to R_2$

$\left[ \begin{array}{cc|cc}
1 & \displaystyle \frac{4}{3} & \displaystyle \frac{1}{3} & 0 \\
0 & \displaystyle \frac{-1}{3} & \displaystyle \frac{-7}{3} & 1
\end{array} \right]$

$\displaystyle -3 R_2$

$\left[ \begin{array}{cc|cc}
1 & \displaystyle \frac{4}{3} & \displaystyle \frac{1}{3} & 0 \\
0 & 1 & 7 & -3
\end{array} \right]$

$\displaystyle R_1 - \frac{4}{3} R_2 \to R_1$

$\left[ \begin{array}{cc|cc}
1 & 0 & -9 & 4 \\
0 & 1 & 7 & -3
\end{array} \right]$

The inverse matrix can now be found in the right half of our reduced row-echelon matrix. So the inverse matrix is

$\left[ \begin{array}{cc}
-9 & 4 \\
7 & -3
\end{array} \right]$