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

⎛
R 1
R 2
→R 2
−6R 1
⎝
R 3
→R 3
−4R 1
⎞
−18−2 100
01 2−610⎠
02 3−401
8.10 Gaussian elimination 293
Thishasremovedalltheoff-diagonalentriesincolumn1.Toremovethoseincolumn2:
⎛
⎞
R 1
→R 1
−8R 2
−10−18 49−80
R 2
⎝ 01 2−6 10⎠
R 3
→R 3
−2R 2
00 −1 8−21
To remove those incolumn 3:
⎛
⎞
R 1
→R 1
−18R 3
−1 0 0 −95 28 −18
R 2
→R 2
+2R 3
⎝ 01 0 10−3 2⎠
R 3
00−1 8−2 1
We mustnow adjust the ‘−1’ entries toobtain the identitymatrix:
⎛
⎞
R 1
→ −R 1
100 95−28 18
R 2
⎝ 010 10 −3 2⎠
R 3
→ −R 3
001−8 2−1
Finally, the required inverse isthe matrix remaining on the r.h.s.:
⎛ ⎞
95 −28 18
⎝ 10 −3 2⎠
−8 2 −1
You should check thisresultby evaluatingAA −1 .
EXERCISES8.10
1 Solvethe followingequations byGaussian
elimination:
(a) 2x−3y=32
3x+7y=−21
(b) 2x+y−3z=−5
x−y+2z=12
7x−2y+3z=37
(c) x+y−z=1
3x−y+5z=3
7x+2y+3z=7
(d) 2x+y−z=−9
3x−2y+4z=5
−2x−y+7z=33
(e) 4x+7y+8z=2
5x+8y+13z=0
3x+5y+7z=1
2 Use Gaussian elimination to solve
x+y+z=7
x−y+2z=9
2x+y−z=1
3 Find the inversesofthe following matrices usingthe
technique ofExample 8.39:
( ) 4 1
(a)
3 2
⎛ ⎞
421
(b) ⎝ 034⎠
−113
⎛ ⎞
103
(c) ⎝ 215⎠
−721