082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017
276 Chapter 8 Matrix algebra( )a bIf A is the matrix , we write its determinant asc d∣ a bc d∣ . Note that the straightlines || indicate that we are discussing the determinant, which is a scalar, rather thanthe matrix itself. If the matrixAis such that |A| = 0, then it has no inverse and is saidtobe singular. If |A| ≠ 0 thenA −1 exists andAissaidtobe non-singular.Asingular matrixAhas |A| = 0.Anon-singular matrixAhas |A| ≠ 0.Example8.23 IfA =( ) 1 2andB=5 0( ) −1 2find |A|, |B|and |AB|.−3 1Solution |A| =∣ 1 25 0∣ = (1)(0) − (2)(5) = −10|B| =∣ −1 2−3 1∣ = (−1)(1) − (2)(−3) =5( )( ) ( )1 2 −1 2 −7 4AB ==5 0 −3 1 −5 10|AB| = (−7)(10) − (4)(−5) = −50We note that |A||B| = |AB|.The resultobtained inExample 8.23 istruemore generally:IfAandBaresquare matrices of the same order, |A||B| = |AB|.8.6.2 OrthogonalmatricesA non-singular square matrixAsuch thatA T = A −1 is said to be orthogonal. Consequently,ifAisorthogonalAA T =A T A =I.Example8.24 Find the inverse ofA =( ) 0 −1. Deduce thatAisan orthogonal matrix.1 0Solution From the formula forthe inverse of a 2 ×2 matrix wefindA −1 = 1 ( ) ( ) 0 1 0 1=1 −1 0 −1 0This isclearly equal tothe transpose ofA. HenceAisan orthogonal matrix.To findthe inversesoflargermatricesweshallneedtostudydeterminantsfurther.Thisisdone inSection8.7.
8.6 The inverse of a 2 × 2 matrix 277EXERCISES8.61 IfA=( ) 5 6findA−4 8−1 .2 Find the inverse, ifitexists,ofeach ofthe followingmatrices:( ) ( ) ( )1 0 −1 0 2 3(a) (b) (c)0 1 0 −1 4 1( ) ( ) ( )−1 0 6 2 −6 2(d) (e) (f)−1 7 9 3 9 3⎛ ⎞1 1(g) ⎜22⎝0 1 ⎟⎠2( ) ( ) 3 0 7 83 IfA= andB=−1 4 4 3find |AB|, |BA|.( ) a b4 IfA= ,B =c dfindAB, |A|, |B|, |AB|.( ) e fg hVerifythat |AB| = |A||B|.( ) 1 25 IfA= findA3 4−1 .Find values ofthe constantsaandbsuch thatA +aA −1 =bI.( ) ( )1 1 2 16 IfA= andB=0 3 −1 3findAB, (AB) −1 ,B −1 ,A −1 andB −1 A −1 .Deduce that (AB) −1 =B −1 A −1 .7 Given that the matrix⎛⎞cos ωt −sin ωt 0M = ⎝sin ωt cos ωt 0⎠0 0 1is orthogonal, findM −1 .( ) a b8 (a) IfA= andkis ascalar constant,c dshowthat the inverse ofthe matrixkAis 1 k A−1 .( ) 1 1(b) Findthe inverse of andhence write1 0⎛ ⎞1 1down the inverse of ⎜33⎟⎝13 0 ⎠ .Solutions164( ) 8 −64 51⎛( ) ( ) 1 0 −1 0− 1 ⎞32 (a) (b) (c) ⎜ 10 100 1 0 −1 ⎝ 2− 1 ⎟⎠5 5⎛( ) −1 0(d)− 1 − 1 ⎞11 (e) No inverse (f) ⎜ 12 18⎟⎝ 1 1 ⎠7 7( )4 62 −2(g)0 23 −132, −1324( ) ae +bg af+bh,ce+dg cf+dhad−bc,eh−fg,(ad −bc)(eh − fg)( ) −2 15 32 −1 a=−2,b=52( ) 1 46 AB= , (AB)−3 9−1 = 1 ( ) 9 −4,21 3 1B −1 = 1 ( ) 3 −1,A7 1 2−1 = 1 ( ) 3 −13 0 1⎛⎞cos ωt sinωt 07 ⎝−sin ωt cos ωt 0⎠0 0 1( ) ( ) 0 1 0 38 (b) ,1 −1 3 −3
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8.6 The inverse of a 2 × 2 matrix 277
EXERCISES8.6
1 IfA=
( ) 5 6
findA
−4 8
−1 .
2 Find the inverse, ifitexists,ofeach ofthe following
matrices:
( ) ( ) ( )
1 0 −1 0 2 3
(a) (b) (c)
0 1 0 −1 4 1
( ) ( ) ( )
−1 0 6 2 −6 2
(d) (e) (f)
−1 7 9 3 9 3
⎛ ⎞
1 1
(g) ⎜2
2
⎝
0 1 ⎟
⎠
2
( ) ( ) 3 0 7 8
3 IfA= andB=
−1 4 4 3
find |AB|, |BA|.
( ) a b
4 IfA= ,B =
c d
findAB, |A|, |B|, |AB|.
( ) e f
g h
Verifythat |AB| = |A||B|.
( ) 1 2
5 IfA= findA
3 4
−1 .
Find values ofthe constantsaandb
such thatA +aA −1 =bI.
( ) ( )
1 1 2 1
6 IfA= andB=
0 3 −1 3
findAB, (AB) −1 ,B −1 ,A −1 andB −1 A −1 .
Deduce that (AB) −1 =B −1 A −1 .
7 Given that the matrix
⎛
⎞
cos ωt −sin ωt 0
M = ⎝sin ωt cos ωt 0⎠
0 0 1
is orthogonal, findM −1 .
( ) a b
8 (a) IfA= andkis ascalar constant,
c d
showthat the inverse ofthe matrixkA
is 1 k A−1 .
( ) 1 1
(b) Findthe inverse of andhence write
1 0
⎛ ⎞
1 1
down the inverse of ⎜3
3
⎟
⎝1
3 0 ⎠ .
Solutions
1
64
( ) 8 −6
4 5
1
⎛
( ) ( ) 1 0 −1 0
− 1 ⎞
3
2 (a) (b) (c) ⎜ 10 10
0 1 0 −1 ⎝ 2
− 1 ⎟
⎠
5 5
⎛
( ) −1 0
(d)
− 1 − 1 ⎞
1
1 (e) No inverse (f) ⎜ 12 18
⎟
⎝ 1 1 ⎠
7 7
( )
4 6
2 −2
(g)
0 2
3 −132, −132
4
( ) ae +bg af+bh
,
ce+dg cf+dh
ad−bc,eh−fg,
(ad −bc)(eh − fg)
( ) −2 1
5 3
2 −1 a=−2,b=5
2
( ) 1 4
6 AB= , (AB)
−3 9
−1 = 1 ( ) 9 −4
,
21 3 1
B −1 = 1 ( ) 3 −1
,A
7 1 2
−1 = 1 ( ) 3 −1
3 0 1
⎛
⎞
cos ωt sinωt 0
7 ⎝−sin ωt cos ωt 0⎠
0 0 1
( ) ( ) 0 1 0 3
8 (b) ,
1 −1 3 −3
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