Page 170 - 35Linear Algebra

170 Determinants

Figure 8.1: Memorize the determinant formula for a 2×2 matrix!

2
1
2
1
m m − m m 6= 0 .
1
2
1
2
For 2 × 2 matrices, this quantity is called the determinant of M.
1 1
m m
1
2
2
1
det M = det 1 2 = m m − m m .
m 2 m 2 1 2 2 1
1 2
Example 101 For a 3 × 3 matrix,
 1 1 1 
m m m
1 2 3
2
M = m 2 1 m 2 m  ,


2
3
m 3 1 m 3 m 3 3
2
then—see review question 1—M is non-singular if and only if:
2
3
1
3
2
1
2
1
3
1
1
2
3
3
3
2
1
2
det M = m m m − m m m + m m m − m m m + m m m − m m m 6= 0.
1 2 3 1 3 2 2 3 1 2 1 3 3 1 2 3 2 1
Notice that in the subscripts, each ordering of the numbers 1, 2, and 3 occurs exactly
once. Each of these is a permutation of the set {1, 2, 3}.
8.1.2 Permutations
Consider n objects labeled 1 through n and shuffle them. Each possible shuf-
fle is called a permutation. For example, here is an example of a permutation
of 1–5:

1 2 3 4 5
σ =
4 2 5 1 3
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