18.3 Matrix algebra 509
Properties of matrix multiplication
The associative law
A(BC) 1 = 1 (AB)C 1 = 1 ABC (18.29)
Matrix multiplication is associative, and the brackets can be omitted.
The distributive law
A(B 1 + 1 C) 1 = 1 AB 1 + 1 AC (18.30)
The commutative law
Matrix multiplication is non-commutativein general (but not in every case),
3AB 1 ≠ 1 BA (18.31)
If Ais an m 1 × 1 nmatrix and Bis an n 1 × 1 pmatrix then ABis m 1 × 1 pbut BAis not
defined unlessp 1 = 1 m, in which caseABis a square matrix of order m, andBAis a
square matrix of order n. Cases (iii) and (iv) in Examples 18.8 demonstrate the
non-commutation for the product of a row matrix and a column matrix. In this case
ABandBAhave different dimensions. Only square matrices maycommute.
EXAMPLE 18.9Commuting matrices
An example of a pair of commutingmatrices is
for which
0 Exercise 38
AB=
=
=
01
10
21
12
12
21
21
112
01
10
=BA
AB=
,=
01
10
21
12
3Hamilton’s algebra of quaternions and Cayley’s matrix algebra were the first examples of algebras not
restricted by the commutative law of multiplication. They led to the development of general algebras that has
continued to this day.