College Algebra, Chapter 7, 7.4, Section 7.4, Problem 26

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

Let

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

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

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

$\displaystyle \det (A) = -1 \left[ 1 (0 \cdot 0 - 4 \cdot 2) - 0 (3 \cdot 0 - 4 \cdot 1) + 2 (3 \cdot 2 - 1 \cdot 0) \right] + 6 \left[ 1 (-4 \cdot 0 - 4 \cdot 0) - 2 (3 \cdot 0 - 4 \cdot 1) + 2 (3 \cdot 0 - (-4) \cdot 1) \right]$

$\displaystyle \det (A) = -4 + 96$

$\displaystyle \det (A) = 92$

The given matrix has an inverse.