MATRICES AND VECTOR SPACES
These are clearly not the same, sincePis anM×Mmatrix whilstQis an
N×Nmatrix. Thus, particular care must be taken to write matrix products in
the intended order;P=ABbutQ=BA. We note in passing thatA^2 meansAA,
A^3 meansA(AA)=(AA)Aetc. Even if bothAandBare square, in general
AB=BA, (8.35)i.e. the multiplication of matrices is not, in general, commutative.
EvaluateP=ABandQ=BAwhereA=
32 − 1
03 2
1 − 34
, B=
2 − 23
110
321
.
As we saw for the 2×2 case above, the elementPijof the matrixP=ABis found by
mentally taking the ‘scalar product’ of theith row ofAwith thejth column ofB.For
example,P 11 =3×2+2×1+(−1)×3=5,P 12 =3×(−2) + 2×1+(−1)×2=−6, etc.
Thus
P=AB=
32 − 1
03 2
1 − 34
2 − 23
110
321
=
5 − 68
972
1137
,
and, similarly,
Q=BA=
2 − 23
110
321
32 − 1
03 2
1 − 34
=
9 −11 6
351
1095
.
These results illustrate that, in general, two matrices do not commute.
The property that matrix multiplication is distributive over addition, i.e. that(A+B)C=AC+BC (8.36)and
C(A+B)=CA+CB, (8.37)follows directly from its definition.
8.4.3 The null and identity matricesBoth the null matrix and the identity matrix are frequently encountered, and we
take this opportunity to introduce them briefly, leaving their uses until later. The
nullorzeromatrix 0 has all elements equal to zero, and so its properties are
A0= 0 =0A,A+ 0 = 0 +A=A.