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

8.9 Application to the solution of simultaneous equations 285
Example8.32 Express the following equations inthe formAX =Band hence solve them:
3x+2y−z =4
2x−y+2z =10
x−3y−4z =5
Solution Usingthe rules of matrix multiplication, wefind
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
3 2−1 x 4
⎝2−1 2⎠
⎝y⎠ = ⎝10⎠
1−3−4 z 5
whichisintheformAX =B.ThematrixAiscalledthecoefficientmatrixandissimply
the coefficients ofx,yandzinthe equations. As before,
AX=B
A −1 AX = A −1 B
IX =X=A −1 B
We musttherefore find the inverse ofAinorder tosolve the equations.
To invertAweuse the adjoint. If
⎛ ⎞
3 2−1
A = ⎝2−1 2⎠
1−3−4
then
⎛ ⎞
3 2 1
A T = ⎝ 2−1−3⎠
−1 2−4
and you should verify thatadj(A) isgiven by
⎛ ⎞
10 11 3
adj(A) = ⎝ 10 −11 −8⎠
−5 11 −7
The determinant ofAis found by expanding along the first row:
|A|=3
∣ −1 2
∣ ∣ ∣∣∣ −3 −4∣ −2 2 2
∣∣∣ 1 −4∣ −1 2 −1
1 −3∣
Therefore,
= (3)(10) − (2)(−10) − (1)(−5)
=30+20+5
= 55
A −1 = adj(A)
|A|
⎛ ⎞
= 1 10 11 3
⎝ 10 −11 −8⎠
55
−5 11 −7