Editing Determinant (section)

===Immediate consequences===
These rules have several further consequences:
* The determinant is a [[homogeneous function]], i.e., <math display="block">\det(cA) = c^n\det(A)</math> (for an <math>n \times n</math> matrix <math>A</math>).
* Interchanging any pair of columns of a matrix multiplies its determinant by&nbsp;−1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): <math display="block">|a_1, \dots, a_j, \dots a_i, \dots, a_n| = - |a_1, \dots, a_i, \dots, a_j, \dots, a_n|.</math> This formula can be applied iteratively when several columns are swapped. For example <math display="block">|a_3, a_1, a_2, a_4 \dots, a_n| = - |a_1, a_3, a_2, a_4, \dots, a_n| = |a_1, a_2, a_3, a_4, \dots, a_n|.</math> Yet more generally, any permutation of the columns multiplies the determinant by the [[parity of a permutation|sign]] of the permutation.
* If some column can be expressed as a linear combination of the ''other'' columns (i.e. the columns of the matrix form a [[Linearly independent|linearly dependent]] set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
* Adding a scalar multiple of one column to ''another'' column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
* If <math>A</math> is a [[triangular matrix]], i.e. <math>a_{ij}=0</math>, whenever <math>i>j</math> or, alternatively, whenever <math>i<j</math>, then its determinant equals the product of the diagonal entries: <math display="block">\det(A) = a_{11} a_{22} \cdots a_{nn} = \prod_{i=1}^n a_{ii}.</math> Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a [[diagonal matrix]] (without changing the determinant).  For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation <math>\sigma</math> which gives a non-zero contribution is the identity permutation.

====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>