Matrices (Chapter 12) 317
Pn
r=1
means the
sum from r=1
to r=n.
Lisa’stotal costat Store A is $ 12 : 30 , and at store B is $ 11 : 20.
Olu’stotal costat Store A is 1 £$ 2 :65 + 2£$ 1 :55 + 2£$ 2 :35 =$ 10 : 45 ,
and at Store B is 1 £$ 2 :25 + 2£$ 1 :50 + 2£$ 2 :20 =$ 9 : 65.
So, using matrices we require that
Having observed the usefulness of multiplying matrices in the contextual examples above, we now define
matrix multiplication more formally.
Theproductof an m£n matrixAwith an n£p matrixB, is the
m£p matrixABin which the element in therth row andcth column
is the sum of the products of the elements in therth row ofAwith the
corresponding elements in thecth column ofB.
If C=AB then cij=
Pn
r=1
airbrj=ai 1 b 1 j+ai 2 b 2 j+::::+ainbnj
for each pairiandjwith 16 i 6 m and 16 j 6 p.
Note that the productABexistsonlyif the number of columns ofA
equals the number of rows ofB.
For example:
If A=
μ
ab
cd
¶
and B=
μ
pq
rs
¶
, then AB=
μ
ap+br aq+bs
cp+dr cq+ds
¶
.
If C=
μ
abc
def
¶
and D=
0
@
x
y
z
1
A, then CD=
μ
ax+by+cz
dx+ey+fz
¶
.
To get the matrixAByou multiplyrows by columns. To get the element in the 5 th row and 3 rd column of
AB(if it exists), multiply the 5 th row ofAby the 3 rd column ofB.
2 £ 3
3 £ 1
2 £ 1
row 1 £column 1
row 1 £column 2
μ
231
122
¶
£
0
@
2 :65 2: 25
1 :55 1: 50
2 :35 2: 20
1
A =
μ
12 :30 11: 20
10 :45 9: 65
¶
row 2 £column 2
row 2 £column 1
2 £ 3 the same 3 £ 2
resultant matrix
2 £ 2
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Y:\HAESE\CAM4037\CamAdd_12\317CamAdd_12.cdr Wednesday, 8 January 2014 9:48:05 AM BRIAN