Determinant

Determinant


	Determinant

In order to be able to use Cramer's rule we need to know how to calculate determinant

$\begin{displaymath} \det(a_1 \; a_2 \; \dots a_n). \end{displaymath}$

The definition for determinant of a matrix is based on some properties of permutations. Consider a set of

elements. A one-to-one mapping of this set into itself is cold a permutation (of order n. The set of all permutations of order n is denoted by

and called symmetric group. See

http://en.wikipedia.org/wiki/Symmetric_group

for details.

Example 1 Symmetric group
$S_2$

has two elements

$\begin{displaymath} (1\;2),\;\;\;(2\;1) \end{displaymath}$

Symmetric group

has six elements

$\begin{displaymath} (1\;2\;3),\;\;(1\;3\;2),\;\;(2\;1\;3),\;\;(2\;3\;1),\;\;(3\;1\;2),\;\;(3\;2\;1) \end{displaymath}$

It is not difficult to see that

has $n!$ elements, where

$n!=1\cdot 2 \dots n.$

We can calculate the number of inversions in each permutations. Inversion is when a larger number precedes the smaller. For example, permutation

$\begin{displaymath} (2\;1\;3) \end{displaymath}$

has one inversion because 2 is placed before1. On the other hand,

$\begin{displaymath} (2\;3\;1) \end{displaymath}$

has two inversions: 2 before 1, and 3 before 1.

Now we can define the signature of a permutation

$\sigma \in S_n:$

$\begin{displaymath} sgn(\sigma) = \left\{ \begin{array}{cc} +1, & \mbox{ if } \... ...f inversions}\ -1, &\mbox{ otherwise } \end{array} \right. \end{displaymath}$

Definition 1

The determinant of a square matrix

$A\in {\rm R}^{n\times n}$ ,

$\begin{displaymath} A=\left( \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1... ... a_{n1} & a_{n2} & \dots & a_{nn} \ \end{array} \right), \end{displaymath}$

is defined as

$\begin{displaymath} \det(A) = \sum_{\sigma \in S_n } sgn(\sigma) \cdot a_{1 \sigma(1)}\cdot a_{2 \sigma(2)}\dots a_{n \sigma(n)} \end{displaymath}$

Given

$A\in {\rm R}^{n\times n}$

calculate

$\det(A)$

is not an easy exercise. However, if

is a triangular matrix, then

$\begin{displaymath} det(A) = a_{11} \cdot a_{22} \cdot a_{33} \dots a_{nn} \end{displaymath}$

On the other hand, elementary row (column operations) can be used to bring a matrix into a triangular form. It is important to notice how the determinant is changing when the matrix undergoes elementary row (column operations).

i.: $\det(A)$ changes its sign when two neighboring rows (or columns) are exchanged.
ii.: $\det(A)$ is multiplied by a number $\lambda$ when a row (or column) is multiplied by $\lambda .$
iii.: $\det(A)$ does not change its value when any row (column) is replaced by its sum with multiple of any other row (column).

In terms of columns the elementary operations can be presented as follows.

i.: $\det(a_1\;a_2\;\dots a_i\;a_{i+1}\;\dots a_n) = - \det(a_1\;a_2\;\dots a_{i+1}\; a_i \;\dots a_n)$
ii.: $\det(a_1\;a_2\;\dots \lambda \cdot a_i\; \dots a_n) = \lambda \cdot \det(a_1\;a_2\;\dots a_i\; \dots a_n)$
iii.: $\det(a_1\;a_2\;\dots a_i\; \dots a_j\;\dots a_n)=\det(a_1\;a_2\;\dots a_i + \lambda \cdot a_j\; \dots a_j\;\dots a_n)$

Hence, we have the following recipe for calculating

$\det(A).$

Use both elementary row and column operations in order to bring

into a triangular form and then calculate the determinant as the product of diagonal elements.

There are many different ways to calculate

$\det(A).$

We mention here only one recursive method which follows immediately from the definition of

$\det(A).$

$\begin{displaymath} \det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \det(M_{ij}), \end{displaymath}$

where the matrix

$M_{ij}$ is obtained from $A$

by crossing out ith row and jth column.

Next: Exercises Up: Cramer's rule Previous: Linear systems with two unknowns

Sergey Nikitin 2004-09-09