082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.3 Addition, subtraction and multiplication 261
Ingeneral wehave
⎛
⎞ ⎛
⎞
a 11
a 12
... a 1n
ka 11
ka 12
... ka 1n
a 21
a 22
... a 2n
IfA= ⎜
⎝
.
.
. ..
. ..
⎟
⎠ thenkA = ka 21
ka 22
... ka 2n
⎜
⎝
.
.
. ..
. ..
⎟
⎠
a m1
a m2
... a mn
ka m1
ka m2
... ka mn
8.3.3 Matrixmultiplication
Matrix multiplication is defined in a special way which at first seems strange but is in
factveryuseful.IfAisa p ×qmatrixandBisanr×smatrixwecanformtheproduct
AB only ifq =r; that is, only if the number of columns inAis the same as the number
of rows inB. The product isthen a p ×smatrixC, thatis
C=AB where Aisp×q
Bisq×s
Cisp×s
Example8.5 GivenA = ( 4 2 ) andB=
Solution Ahas size1 ×2
Bhassize2×3
( ) 3 7 6
can the productAB beformed?
52 −1
Because the number of columns inAis the same as the number of rows in B, we can
formtheproductAB.Theresultingmatrixwillhavesize1 ×3becausethereisonerow
inAand three columns inB.
SupposewewishtofindABwhenA = ( 4 2 ) andB=
( 3
7)
.Ahassize1 ×2andBhas
size2 ×1andsowecanformtheproductAB. Theresultwillbea1 ×1matrix,thatis
a singlenumber. We perform the calculation as follows:
AB = ( 4 2 )( )
3
=4×3+2×7=12+14=26
7
Note that we have multiplied elements in the row ofAwith corresponding elements in
the column ofB, and added the results together.
⎛ ⎞
Example8.6 FindCD whenC = ( 192 ) 2
andD= ⎝6⎠.
8
⎛ ⎞
Solution CD = ( 192 ) 2
⎝6⎠ =1×2+9×6+2×8=2+54+16=72
8
Let us now extend this idea to general matrices A and B. Suppose we wish to findC
whereC =AB.Theelementc 11
isfoundbypairingeachelementinrow1ofAwiththe