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

8.8 The inverse of a 3 × 3 matrix 281
EXERCISES8.7
∣ ∣ 1 Find
∣ 4 6
∣∣∣∣∣ 13 4
∣∣∣∣∣ 2 8∣ , 672
21 0
35 −1∣ and 143
−114∣ .
2 Find
cos ωt sin ωt
∣−sin ωt cos ωt∣ .
∣ 500
∣∣∣∣∣ 900
3 Evaluate
632
∣457∣ and 070
008∣ .
⎛ ⎞
2 −17
4 IfA= ⎝0 84⎠,find |A|and |A T |.
3 64
Comment uponyour result.
5 UseCramer’s ruleto solve
(a) 2x−3y+z=0
5x+4y+z=10
2x−2y−z=−1
(b) 3x+y=−1
2x−y+z=−1
5x+5y−7z=−16
(c) 4x+y+z=13
2x−y=4
x+y−z=−3
(d) 3x+2y=1
x+y−z=1
2x+3z=−1
6 Given
⎛ ⎞
37 6
A = ⎝−21 0⎠
42 −5
(a) find |A|
(b) findthe cofactors ofthe elements ofrow 2,that is
−2,1, 0
(c) calculate
−2 × (cofactorof −2)
+1 × (cofactorof1)
+0 × (cofactorof0).
What doyou deduce?
7 Ifa=7i+11j−2kandb=6i−3j+kfinda×b.
8 Find a ×bwhen
(a) a=3i−j,b=i+j+k
(b) a=2i+j+k,b=7k
(c) a=−7j−k,b=−3i+j
Solutions
1 20,33,39
2 1
3 55,504
4 −164, −164 Note |A| = |A T |
5 (a) x=y=z=1
(b) x=−1,y=2,z=3
(c) x=2,y=0,z=5
(d) x=1,y=−1,z=−1
6 (a) −133 (b) 47, −39,22 (c) −133
7 5i−19j−87k
8 (a) −i−3j+4k
(b) 7i −14j
(c) i+3j−21k
8.8 THEINVERSEOFA3×3MATRIX
Given a 3×3 matrix,A, its inverse isfound as follows:
(1) Find the transpose ofA, by interchanging the rows and columns ofA.
(2) ReplaceeachelementofA T byitscofactor;byitsminortogetherwithitsassociated
place sign. The resulting matrix isknown as theadjoint ofA, denoted adj(A).