Page 545 - Modelling in Transport Phenomena A Conceptual Approach

A.9. MATRICES 525

A.9.2 Determinants
For each square matrix A, it is possible to associate a scalar quantity called the
determinant of A, IAl. If the matrix A in &. (A.9-1) is a square matrix, then the
determinant of A is given by

all a12 a13 aln
a21 a22 a23 a2n
PI = . , . . . . . . . . . . . . . , . . . . . . . (A.9-13)
an1 an2 an3 ann
If the row and column containing an element aij in a square matrix A are deleted,
the determinant of the remaining square array is called the minor of aij and denoted
by Mij. The cofactor of aij, denoted by A,, is then defined by the relation

-
A.. - (- l)i+iMij (A.9-14)
'3
Thus, if the sum of the row and column indices of an element is even, the cofactor
and the minor of that element are identical; otherwise they differ in sign.
The determinant of a square matrix A can be calculated by the following for-
mula:
n n
(A.9- 15)
k=l k=l
where i and j may stand for any row and column, respectively. Therefore, the
determinant of 2 x 2 and 3 x 3 matrices are

all a12
= %la22 - 012021 (A.9-16)
a21 a22
all a12 a13
a21 a22 a23
a31 a32 a33

Example A.ll Determine IAI if
1 0 1
A=[ 3 2 11
-1 1 0

Solution
Expanding on the first row, i.e., i = 1, gives