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

292 Chapter 8 Matrix algebra
ThecoefficientmatrixisknownasaVandermondematrix.Supposewewishtofind
asecond-degreepolynomialapproximationtothesignal f (t) = cos πt
2 forvaluesof
t between −1 and 1. We can do this by making the approximating polynomial and
the original signal equal at three points, sayt = −1,t = 0 andt = 1. The equations
tobe solved arethen
⎛ ⎞⎛
⎞ ⎛ ⎞ ⎛ ⎞
1−11 a 0
f(−1) 0
⎝1 00⎠⎝a 1
⎠ = ⎝ f(0) ⎠ = ⎝1⎠
1 11 a 2
f(1) 0
It is straightforward to solve this system of equations by Gaussian elimination and
obtaina 0
=1,a 1
=0 anda 2
= −1. Therefore the second-degree polynomial which
approximates f (t) = cos πt
2 is1−t2 .
8.10.1 Findingtheinversematrixusingrowoperations
AsimilartechniquecanbeusedtofindtheinverseofasquarematrixAwherethisexists.
Suppose wearegiven the matrixAand wish tofind its inverseB. Thenweknow
AB=I
that is,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
a 11
a 12
a 13
b 11
b 12
b 13
100
⎝a 21
a 22
a 23
⎠ ⎝b 21
b 22
b 23
⎠ = ⎝010⎠
a 31
a 32
a 33
b 31
b 32
b 33
001
We form the augmentedmatrix
⎛
⎞
a 11
a 12
a 13
1 0 0
⎝a 21
a 22
a 23
0 1 0⎠
a 31
a 32
a 33
0 0 1
Nowcarryoutrowoperationsonthismatrixinsuchawaythatthel.h.s.isreducedtoa
3 ×3identitymatrix.Thematrixwhichthenremainsonther.h.s.istherequiredinverse.
Example8.39 Findthe inverse of
⎛ ⎞
−18 −2
A = ⎝−6 49 −10⎠
−4 34 −5
by rowreduction tothe identity.
Solution We form the augmentedmatrix
⎛
⎞
−1 8 −2 100
⎝−649−10 010⎠
−434 −5 001
We now carry out row operations on the whole matrix to reduce the l.h.s. to an identity
matrix. This means we must eliminate all the elements off the diagonal. Work through
the following calculationyourself: