Cambridge Additional Mathematics

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

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_12\317CamAdd_12.cdr Wednesday, 8 January 2014 9:48:05 AM BRIAN