Page 544 - Modelling in Transport Phenomena A Conceptual Approach

524 APPENDIX A. MATHEMATICAL PRELIMINARlES

matrices are said to be conformable in the order stated. For example, if A is
AB = [ all a12 [ bll b12 ]
of order 4 x 2 and B is of order 2 x 3, then the product AB is

bzl bzz b13 (A.9-8)
b23
=[ a4lbll + a4241 a4lbl2 + a4242 a41b13 + a42b23 1
a41
a42
allbll + alZb2l
allbl2 + a12b22 allbl3 + a1243
a2lbll + a22b2l
a21b13 + a22b23
a21bl2 + a22b22
a3lhl + a32b21
a3lbl2 + a32b22 a3lbl3 + a32b23
In general, if a matrix of order (m, is multiplied by a matrix of order (T, n),
r)
the product is a matrix of order (m, n). Symbolically, this may be expressed
as

Example A.10 If
1 -1 and .=[:I
A= 2 0
-1 5
determine AB.
Solution

AB = [ : il][i]

-1 5

6. A matrix A can be multiplied by itself if and only if it is a square matrix. The
product AA can be expressed as A2. If the relevant products are defined,
multiplication of matrices is associative, i.e.,
A(BC) = (AB)C (A.9-10)
and distributive, i.e.,
A(BtC)=AB+AC (A. 9-1 1)
(B+C)A = BA+CA (A.9-12)
but, in general, not commutative.