4.3.3.1 Addition and Subtraction
Matrices of the same size (2&2 and 2&2, 3&1 and 3&1, etc.) can be added just
by adding corresponding elements:
21
74
þ13
56
¼
2 þ 11 þ 3
7 þ 54 þ 6¼
34
12 10
7
0
3
0
B
@
1
C
Aþ4
4
1
0
B
@
1
C
A¼
7 þ 4
0 þ 4
3 þ 10
B
@
1
C
A¼
11
4
4
0
B
@
1
C
A
Subtraction is similar:21
74
"^13
56
¼^2 "^11 "^3
7 " 54 " 6
¼^1 "^2
2 " 2
4.3.3.2 Multiplication by a Scalar
A scalar is an ordinary number (in contrast to a vector or an operator), e.g. 1, 2,
pffi
2,
1.714,p, etc. To multiply a matrix by a scalar we just multiply every element by the
number:
2
21
74
¼
2 & 22 & 1
2 & 72 & 4
¼
42
14 8
4.3.3.3 Matrix Multiplication
We could define matrix multiplication to be analogous to addition: simply
multiplying corresponding elements. After all, in mathematics any rules are
permitted, as long as they do not lead to contradictions. However, as we shall
see in a moment, for matrices to be useful in dealing with simultaneous equations
we must adopt a slightly more complex multiplication rule. The easiest way to
understand matrix multiplication is to first define series multiplication. If series
a¼Sa¼a 1 a 2 a 3 ..., and seriesb¼Sb¼b 1 b 2 b 3 ...then we define the series
product as
SaSb¼a 1 b 1 þa 2 b 2 þa 3 b 3 þ:::110 4 Introduction to Quantum Mechanics in Computational Chemistry