Page 45 - Numerical Methods for Chemical Engineering

34 1 Linear algebra

Expansion by minors

From (1.168), it may be shown that det(A) may be written as an expansion in minors along
any row j or column j as

N N
j+k
det(A) = a jk C jk = a kj C kj C jk = (−1) M jk (1.171)
k=1 k=1
M jk (the minor of a jk ) is the determinant of the (N − 1)×(N − 1) matrix obtained by
deleting row j and column k of A. The quantity C jk is the cofactor of a jk .

The determinant of 2 × 2 and 3 × 3 matrices
Expansion of minors can be useful when computing the determinant of a matrix, as the
minors are determinants of smaller matrices that are easier to compute. For example, the
determinant of a 3 × 3 matrix can be written as

a 22 a 23 a 21 a 23 a 21 a 22
(1.172)

a 32 a 33 − a 12 a 31 a 33 + a 13 a 31 a 32
det(A) = a 11
As the determinant of a 2 × 2 matrix is

a 11 a 12
det(A) = = ε 12 a 11 a 22 + ε 21 a 12 a 21 = a 11 a 22 − a 12 a 21 (1.173)
a 21 a 22

expansion by minors provides an easy means to compute the determinants of small matrices.
A general numerical approach is discussed below.

General properties of the determinant function

We now consider some general properties of the determinant. The proofs for some are given
in the supplemental material in the accompanying website.

Property I
T
The determinant of an N × N real matrix A equals that of its transpose, A .

Property II
If every element in a row (column) of A is zero, det(A) = 0.

Property III
If every element in a row (column) of a matrix A is multiplied by a scalar c to form a matrix
B, then det(B) = c × det(A).