Revision as of 14:00, 25 April 2011

MA 265 Chapter 3 Sections 3.1-3.2

By: Daniel Ford

What are determinants?

To understand determinants, you must first know about permutations.

Permutations

If D = {1, 2,....,n} a set of integers from 1 to n in ascending order, then a permutation is the rearrangement of an integer in D.

Example:

If D = {6, 7, 8, 9}, then 7689 would be a permutation of D.  This corresponds to the function f: D→ D defined by

        f(1) = 6
        f(2) = 7
        f(3) = 8
        f(4) = 9
Then after permutation,
        f(1) = 7
        f(2) = 6
        f(3) = 8
        f(4) = 9

The reason that this is important to know is that the total number of permutations can be even or odd.  So in the permutation 4321, where 4 precedes 3, 4 precedes 1, 4 prececedes 2, 3 precedes 1, and 3 precedes 2.  Thus the total number of inversions in this premutation is 5, which would make 4321, odd.  This determines what sign you put in front, so if it is odd, then you put a negative sign (-) in front; and if it is even, then you put a postive sign (+) in front.

Determinants

Now that you know what permutations are, we can start to talk about determinants.

Let A = [a_ij ] be an n x n matrix.  The determinant function, denoted by det, is defined by

                    det(A) = ∑(±)a_1j1a_2j2•••a_njn

where the summation is over all permutations j₁j₂•••j_n of the set D = {1,2,...,n}.  The sign is taken as positive (+) or negative (-) according to whether the permutation j₁j₂•••j_n is ever or odd.

Example:

If A = [a₁₁] is a 1 x 1 matrix, then det(A) = a₁₁.

Example:

If

  $  A =          \begin{bmatrix}         a<sub>11</sub> & a<sub>12</sub>\\         a<sub>21</sub> & a<sub>22</sub> \end{bmatrix} $

then

  det(A) = a11a22 - a12a21

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