Page 172 - 35Linear Algebra

172 Determinants

Permutation Example

Reading homework: problem 1

We can use permutations to give a definition of the determinant.

Definition The determinant of n × n matrix M is

P
det M = sgn(σ) m 1 m 2 · · · m n .
σ(1) σ(2) σ(n)
σ

The sum is over all permutations of n objects; a sum over the all elements
of {σ : {1, . . . , n} → {1, . . . , n}}. Each summand is a product of n entries
from the matrix with each factor from a different row. In different terms of
the sum the column numbers are shuffled by different permutations σ.
The last statement about the summands yields a nice property of the
determinant:

i
Theorem 8.1.1. If M = (m ) has a row consisting entirely of zeros, then
j
m i = 0 for every σ and some i. Moreover det M = 0.
σ(i)
Example 102 Because there are many permutations of n, writing the determinant
this way for a general matrix gives a very long sum. For n = 4, there are 24 = 4!
permutations, and for n = 5, there are already 120 = 5! permutations.

 1 1 1 1 
m m m m
1 2 3 4
m m m m
 2 2 2 2
, then det M is:
 1 2 3 4
For a 4 × 4 matrix, M =  3 3 3 3
m m m
4
1 2 3 m 
m 4 m 4 m 4 m 4
1 2 3 4
2
3
3
4
4
2
1
1
3
1
2
det M = m m m m − m m m m − m m m m 4 3
4
1
1
2
2
1
2
4
3
4
3
1
4
3
2
2
3
1
3
1
4
2
− m m m m + m m m m + m m m m 4 3
4
2
1
4
2
3
1
4
1
2
3
3
4
2
2
4
1
3
1
+ m m m m + m m m m ± 16 more terms.
3
1
2
4
1
2
4
3
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