Page 188 - 35Linear Algebra

188 Determinants

Example 105

     
1 2 3 1 2 3 1 2 3
det  4 5 6  = det  3 3 3  = det  3 3 3  = 0 .
7 8 9 6 6 6 0 0 0

Try to determine which row operations we made at each step of this computation.

You might suspect that determinants have similar properties with respect
to columns as what applies to rows:

T
If M is a square matrix then det M = det M .

Proof. By definition,

X
det M = sgn(σ)m 1 σ(1) m 2 · · · m n .
σ(2)
σ(n)
σ
For any permutation σ, there is a unique inverse permutation σ −1 that
undoes σ. If σ sends i → j, then σ −1 sends j → i. In the two-line notation
for a permutation, this corresponds to just flipping the permutation over. For

1 2 3
example, if σ = , then we can find σ −1 by flipping the permutation
2 3 1
and then putting the columns in order:

2 3 1 1 2 3
−1
σ = = .
1 2 3 3 1 2
Since any permutation can be built up by transpositions, one can also find
the inverse of a permutation σ by undoing each of the transpositions used to
build up σ; this shows that one can use the same number of transpositions
−1
−1
to build σ and σ . In particular, sgn σ = sgn σ .

Reading homework: problem 5

188