Editing Determinant (section)

====Example====
These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, [[Gaussian elimination]] can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix <math>A</math> using that method:

:<math>A = \begin{bmatrix}
 -2 & -1 & 2 \\
  2 & 1 & 4 \\
 -3 & 3 & -1
\end{bmatrix}. </math>

{| class="wikitable"
|+ Computation of the determinant of matrix <math>A</math>
|-
| Matrix || <math>B = \begin{bmatrix}
  -3 & -1 & 2 \\
  3 & 1 & 4 \\
  0 & 3 & -1
\end{bmatrix} </math> ||
<math>C = \begin{bmatrix}
  -3 & 5 & 2 \\
  3 & 13 & 4 \\
  0 & 0 & -1
\end{bmatrix} </math>
||
<math>D = \begin{bmatrix}
  5 & -3 & 2 \\
 13 & 3 & 4 \\
  0 & 0 & -1
\end{bmatrix} </math>
||
<math>E = \begin{bmatrix}
 18 & -3 & 2 \\
  0 & 3 & 4 \\
  0 & 0 & -1
\end{bmatrix} </math>
|-
| Obtained by ||
add the second column to the first
||
add 3 times the third column to the second
||
swap the first two columns
||
add <math>-\frac{13} 3</math> times the second column to the first
|-
| Determinant || <math>|A| = |B|</math> ||
<math>|B| = |C|</math>
||
<math>|D| = -|C|</math>
||
<math>|E| = |D|</math>
|}

Combining these equalities gives <math>|A| = -|E| = -(18 \cdot 3 \cdot (-1)) = 54.</math>