College Algebra, Chapter 7, 7.4, Section 7.4, Problem 18

Determine the minor $M_{32}$ and co-factor $A_{32}$ using the matrix $\displaystyle A = \left[ \begin{array}{ccc}
1 & 0 & \displaystyle \frac{1}{2} \\
-3 & 5 & 2 \\
0 & 0 & 4
\end{array} \right]$

The minor $M_{32}$ is the determinant of the matrix obtained by deleting the third row and second column from $A$. Thus,

$\displaystyle M_{13} = \left| \begin{array}{ccc}
1 & \color{red}{0} & \displaystyle \frac{1}{2} \\
-3 & \color{red}{5} & 2 \\
\color{red}{0} & \color{red}{0} & \color{red}{4}
\end{array} \right| = \left| \begin{array}{cc}
1 & \displaystyle \frac{1}{2} \\
-3 & 2
\end{array} \right| = \displaystyle \left( \frac{1}{2} \right) (-3) - (1)(2) = \frac{-7}{2}$

So the co-factor $\displaystyle A_{32} = (-1)^{3 + 2} M_{32} = (-1)^5 \left( \frac{-7}{2} \right) = \frac{7}{2}$