Page 186 - 35Linear Algebra

186 Determinants

8.4 Properties of the Determinant

We now know that the determinant of a matrix is non-zero if and only if that
matrix is invertible. We also know that the determinant is a multiplicative
function, in the sense that det(MN) = det M det N. Now we will devise
some methods for calculating the determinant.
Recall that:

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

A minor of an n × n matrix M is the determinant of any square matrix
obtained from M by deleting one row and one column. In particular, any
i
entry m of a square matrix M is associated to a minor obtained by deleting
j
the ith row and jth column of M.
It is possible to write the determinant of a matrix in terms of its minors
as follows:

X
det M = sgn(σ) m 1 m 2 σ(2) · · · m n
σ(1)
σ(n)
σ
X 1 2 n
1
= m
1 sgn(/ σ ) m 1 · · · m 1
/ σ (n)
/ σ (2)
/ σ 1
X
2
n
3
2
+ m 1 2 sgn(/ σ ) m 2 m 2 · · · m 2
/ σ (3)
/ σ (n)
/ σ (1)
/ σ 2
X 3
n
2
4
3
+ m 1 3 sgn(/ σ ) m 3 m 3 m 3 · · · m 3
/ σ (1)
/ σ (2)
/ σ (4)
/ σ (n)
/ σ 3
+ · · ·
k
Here the symbols / σ refers to the permutation σ with the input k removed.
The summand on the j’th line of the above formula looks like the determinant
of the minor obtained by removing the first and j’th column of M. However
j
we still need to replace sum of / σ by a sum over permutations of column
numbers of the matrix entries of this minor. This costs a minus sign whenever
1
j −1 is odd. In other words, to expand by minors we pick an entry m of the
j
first row, then add (−1) j−1 times the determinant of the matrix with row i
and column j deleted. An example will probably help:
186