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

8.3 Addition, subtraction and multiplication 259
wherea ij
representsthenumberorelementintheithrowand jthcolumn.Amatrixwith
a singlecolumn can alsobe regarded asacolumn vector.
Theoperationsofaddition,subtractionandmultiplicationaredefineduponmatrices
and these areexplained inSection 8.3.
8.3 ADDITION,SUBTRACTIONANDMULTIPLICATION
8.3.1 Matrixadditionandsubtraction
Twomatricescanbeadded(orsubtracted)iftheyhavethesameshapeandsize,thatisthe
sameorder.Theirsum(ordifference)isfoundbyadding(orsubtracting)corresponding
elements as the following example shows.
Example8.2 If
A =
( ) 1 5 −2
31 1
and B=
( ) 120
−1 1 4
findA+BandA−B.
Solution
( ) ( ) ( )
1 5 −2 120 2 7 −2
A +B= + =
31 1 −1 1 4 22 5
( ) ( ) ( )
1 5 −2 120 0 3 −2
A−B= − =
31 1 −1 1 4 4 0 −3
Ontheotherhand,thematrices
they have different orders.
( ) ( 1 2 3
and cannotbeaddedorsubtractedbecause
0 1 1)
Example8.3 IfC =
( ) a b
andD=
c d
( ) e f
showthatC+D=D+C.
g h
Solution C+D =
D+C =
( ) ( ) ( )
a b e f a +e b+f
+ =
c d g h c+g d+h
( ) ( ) ( )
e f a b e +a f+b
+ =
g h c d g+c h+d
Nowa +e is exactly the same ase +a because addition of numbers is commutative.
The same observation can be made ofb + f,c +gandd+h. HenceC +D =D +C.
The addition of these matrices is therefore commutative. This may seem an obvious
statementbut weshall shortlymeet matrix multiplication which isnotcommutative, so
ingeneral commutativity should not be simplyassumed.