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

280 Chapter 8 Matrix algebra
Example8.28 Ifa=3i+j−2kandb=4i+5kfinda×b.
Solution We have
ij k
a×b =
31−2
∣40 5∣
=5i−23j−4k
8.7.2 Cramer’srule
A useful application of determinants is to the solution of simultaneous equations. Consider
the case of three simultaneous equations inthreeunknowns:
a 11
x+a 12
y+a 13
z = b 1
a 21
x+a 22
y+a 23
z = b 2
a 31
x+a 32
y+a 33
z = b 3
Cramer’s rulestates thatx,yandzaregiven by the following ratios of determinants.
Cramer’srule:
∣ ∣ ∣ b 1
a 12
a ∣∣∣∣∣
13
a 11
b 1
a ∣∣∣∣∣
13
a 11
a 12
b ∣∣∣∣∣ 1
b 2
a 22
a 23
a 21
b 2
a 23
a 21
a 22
b 2
∣b 3
a 32
a ∣ 33
a
x =
∣ 31
b 3
a ∣ 33
a
y =
∣ 31
a 32
b
z =
3
∣ a 11
a 12
a ∣∣∣∣∣ 13
a 11
a 12
a ∣∣∣∣∣ 13
a 11
a 12
a ∣∣∣∣∣ 13
a 21
a 22
a 23
a 21
a 22
a 23
a 21
a 22
a 23
∣a 31
a 32
a ∣ 33
a 31
a 32
a ∣ 33
a 31
a 32
a 33
Note that in all cases the determinant in the denominator is identical and its elements
arethecoefficientsonthel.h.s.ofthesimultaneousequations.Whenthisdeterminantis
zero, Cramer’s method will clearly fail.
Example8.29 Solve
3x+2y−z =4
2x−y+2z =10
x−3y−4z =5
Solution We find
4 2−1
10−1 2
∣ 5−3−4∣
x =
3 2 −1
2−1 2
∣1−3−4∣
= 165
55 = 3
Verifyfor yourself thaty = −2 andz = 1.