College Algebra, Chapter 7, 7.4, Section 7.4, Problem 24

Determine the determinant of the matrix $\displaystyle \left[ \begin{array}{ccc}
0 & -1 & 0 \\
2 & 6 & 4 \\
1 & 0 & 3
\end{array} \right]$. State whether the matrix has an inverse, but don't calculate the inverse.

Let

$ A = \displaystyle \left[ \begin{array}{ccc}
0 & -1 & 0 \\
2 & 6 & 4 \\
1 & 0 & 3
\end{array} \right]$

$\displaystyle \det (A) = \left[ \begin{array}{ccc}
0 & -1 & 0 \\
2 & 6 & 4 \\
1 & 0 & 3
\end{array} \right] = 0 \left| \begin{array}{cc}
6 & 4 \\
0 & 3
\end{array} \right| - (-1) \left| \begin{array}{cc}
2 & 4 \\
1 & 3
\end{array} \right| + 0 \left| \begin{array}{cc}
2 & 6 \\
1 & 0
\end{array} \right| = 0 (6 \cdot 3 - 4 \cdot 0) - (-1) (2 \cdot 3- 4 \cdot 1) + 0 (2 \cdot 0 - 6 \cdot 1)$

$\det (A) = 2$



The given matrix has an inverse.