Page 268 - Advanced engineering mathematics

248 CHAPTER 8 Determinants

The determinant of an n × n matrix A is defined to be

det A = σ(p)a 1p(1) a 2p(2) ···a np(n) (8.1)
p
with this sum extending over all permutations p of 1,2,··· ,n. Note that det A is a sum of
terms, each of which is plus or minus a product containing one element from each row and
each column of A.

We often denote det A as |A|. This is not to be confused with the absolute value, as a
determinant can be negative.

EXAMPLE 8.1
We will use the definition to evaluate the general 2 × 2 and 3 × 3 determinants. For the 2 × 2
case, we have a matrix

a 11 a 21
A = .
a 21 a 22
There are only two permutations on the numbers 1,2, namely

p 1 : 1,2 → 1,2 and p 2 : 1,2 → 2,1.

It is easy to check that p 1 is even and p 2 is odd. Therefore

|A|= σ(p 1 )a 1p 1 (1) a 2p 1 (2) + σ(p 2 )a 1p 2 (1) a 2p 2 (2)
= a 11 a 22 − a 12 a 21 .
For the 3×3 case, suppose B=[b ij ] is a 3×3 matrix. Now we must use the six permutations
of the integers 1,2,3:

p 1 : 1,2,3 → 1,2,3,(even); p 2 : 1,2,3,→ 1,3,2,(odd);
p 3 : 1,2,3 → 2,3,1,(even); p 4 : 1,2,3,→ 2,1,3,(odd);
p 5 : 1,2,3,→ 3,1,2,(even); p 6 : 1,2,3,→ 3,2,1,(odd).

Then
6

|B|= σ(p k )b 1p k (1) b 2p k (2) b 3p k (3)
k=1
= b 11 b 22 b 33 − b 11 b 23 b 32 + b 12 b 23 b 31
= b 12 b 21 b 33 + b 13 b 21 b 32 − b 13 b 22 b 31 .

There are n!= 1 · 2 · 3···n permutations of 1,2,··· ,n (for example, 120 permutations
of 1,2,3,4,5), so the definition is not a practical method of evaluation. However, it serves
as a starting point to develop the properties of determinants we will need to make use
of them.

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October 14, 2010 14:26 THM/NEIL Page-248 27410_08_ch08_p247-266