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

274 Chapter 8 Matrix algebra
8.6 THEINVERSEOFA2×2MATRIX
Whenwearedealingwithordinarynumbersitisoftennecessarytocarryouttheoperationofdivision.Thus,forexample,ifweknowthat3x
= 4,thenclearlyx = 4/3.Ifwe
are given matricesAandC and know that
AB=C
how do we findB? Itmight be tempting towrite
B = C A
Unfortunately, this would be entirely wrong since division of matrices is not defined.
However, given expressions like AB = C it is often necessary to be able to find the
appropriateexpressionforB.Thisiswhereweneedtointroducetheconceptofaninverse
matrix.
IfAisasquare matrix and we can find another matrixBwith the property that
AB=BA=I
thenBissaid tobe theinverseofAand iswrittenA −1 , thatis
AA −1 =A −1 A =I
IfBis the inverse ofA, thenAis also the inverse ofB. Note thatA −1 does not mean a
reciprocal; there is no such thing as matrix division.A −1 is the notation we use for the
inverse ofA.
Multiplyingamatrix byits inverse yieldsthe identitymatrixI, thatis
AA −1 =A −1 A =I
Example8.19 IfA =
SinceAisasquarematrix,A −1 isalsosquareandofthesameorder,sothattheproducts
AA −1 andA −1 A can be formed. The term ‘inverse’ cannot be applied to a matrix which
isnotsquare.
( ) 2 1
show thatthe matrix
3 2
( ) 2 −1
isthe inverse ofA.
−3 2
Solution Forming the products
( )( ) ( )
2 1 2 −1 1 0
=
3 2 −3 2 0 1
( )( ) ( )
2 −1 2 1 1 0
=
−3 2 3 2 0 1
wesee that
( ) 2 −1
is the inverse ofA.
−3 2