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

286 Chapter 8 Matrix algebra
Finally, the solutionX isgiven by
⎛ ⎞ ⎛ ⎞⎛
⎞
x
X = ⎝y⎠ =A −1 B = 1 10 11 3 4
⎝10 −11 −8⎠⎝10⎠
z
55
−5 11 −7 5
⎛ ⎞
3
= ⎝−2⎠
1
that is,the solution isx = 3,y = −2 andz = 1.
EXERCISES8.9
1 Byexpressing the following equations in matrixform
andfinding an inverse matrix, solve
(a) 4x−2y=14
2x+ y=5
(b) 2x−2y=0
x+3y=−8
(c) 8x+3y=59
−2x+ y=−13
2 Solve the following equationsAX =Bbyfinding
A −1 ,ifitexists.
( ( ) ( ) 6 3 x 12
(a) =
5 2)
y 9
(b)
(c)
(d)
(e)
(f)
( ) ( ) 4 4 x 20
=
1 3)(
y 11
( ) ( 2 −1 x −4
=
3 2)(
y 1)
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
4 13 x 20
⎝2 −14⎠
⎝y⎠ = ⎝20⎠
0 15 z 20
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
4 13 x 15
⎝2 −14⎠
⎝y⎠ = ⎝12⎠
0 15 z 17
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
4 13 x 0
⎝2 −14⎠
⎝y⎠ = ⎝0⎠
0 15 z 0
Solutions
1 (a)x=3,y=−1
(b) x=−2,y=−2
(c)x=7,y=1
2 (a) x=1,y=2 (b) x=2,y=3
(c) x=−1,y=2 (d) x=2,y=0,z=4
(e) x=1,y=2,z=3 (f) x=y=z=0
8.10 GAUSSIANELIMINATION
An alternative technique for the solution of simultaneous equations is that of Gaussian
eliminationwhich weintroduce bymeansofthe following trivialexample.
Example8.33 UseGaussian eliminationtosolve
2x+3y=1
x+y=3