Example 10-2. If A=
(
1 2 3
4 5 6
)
2 × 3
and B=
7 8 9 10
11 12 13 14
15 16 17 18
3 × 4
the ma-
trices Aand Bsatisfy the compatibility condition for matrix multiplication and the
matrix product C =AB will be a matrix having the dimension of 2 rows and 4
columns. One can write
C=
(
c 11 c 12 c 13 c 14
c 21 c 22 c 23 c 24
)
=AB =
(
1 2 3
4 5 6
)
7 8 9 10
11 12 13 14
15 16 17 18
where c 11 is the inner product of row 1 with column 1 giving
c 11 = 1(7) + 2(11) + 3(15) = 74
In a similar fashion one finds
c 12 is the inner product of row 1 with column 2 giving
c 12 = 1(8) + 2(12) + 3(16) = 80
c 13 is the inner product of row 1 with column 3 giving
c 13 = 1(9) + 2(13) + 3(17) = 86
c 14 is the inner product of row 1 with column 4 giving
c 14 = 1(10) + 2(14) + 3(18) = 92
c 21 is the inner product of row 2 with column 1 giving
c 21 = 4(7) + 5(11) + 6(15) = 173
c 22 is the inner product of row 2 with column 2 giving
c 22 = 4(8) + 5(12) + 6(16) = 188
c 23 is the inner product of row 2 with column 3 giving
c 23 = 4(9) + 5(13) + 6(17) = 203
c 24 is the inner product of row 2 with column 4 giving
c 24 = 4(10) + 5(14) + 6(18) = 218
This gives the matrix product
AB =
(
1 2 3
4 5 6
)
7 8 9 10
11 12 13 14
15 16 17 18
=C=
(
74 80 86 92
173 188 303 218
)
Matrices with the proper dimensions satisfy the properties
A(B+C) = AB +AC left distributive law
(B+C)A=BA +CA right distributive law
A(BC ) =(AB )C associative law
