College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 46

Determine the determinant of the matrix $\displaystyle A = \left[
\begin{array}{ccc}
2 & 4 & 0 \\
-1 & 1 & 2 \\
0 & 3 & 2
\end{array}
\right]$ and if possible, the inverse of the matrix.

Using the formula

$\displaystyle |D| = \left[
\begin{array}{ccc}
2 & 4 & 0 \\
-1 & 1 & 2 \\
0 & 3 & 2
\end{array}
\right] = 2 \left|
\begin{array}{cc}
1 & 2 \\
3 & 2
\end{array}
\right| -4 \left|
\begin{array}{cc}
-1 & 2 \\
0 & 2
\end{array}
\right| + 0 \left|
\begin{array}{cc}
-1 & 1 \\
0 & 3
\end{array}
\right| = 2 (1 \cdot 2 - 2 \cdot 3) - 4 (-1 \cdot 2 - 2 \cdot 0) = 0$

Since the determinant of $A$ is zero, $A$ cannot have an inverse, by the invertibility criterion.

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