THE SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRICES AND DETERMINANTS 283F
Find the values of F 1 ,F 2 andF 3 using
determinants.[F 1 =2,F 2 =−3,F 3 =4]- Mesh-current analysis produces the follow-
ing three equations:
20 ∠ 0 ◦=(5+ 3 −j4)I 1 −(3−j4)I 210 ∠ 90 ◦=(3−j 4 +2)I 2 −(3−j4)I 1 − 2 I 3− 15 ∠ 0 ◦− 10 ∠ 90 ◦=(12+2)I 3 − 2 I 2Solve the equations for the loop currents
I 1 ,I 2 andI 3.
[
I 1 = 3. 317 ∠ 22. 57 ◦A
I 2 = 1. 963 ∠ 40. 97 ◦A
I 3 = 1. 010 ∠− 148. 32 ◦A]26.3 Solution of simultaneous
equations using Cramers rule
Cramers rule states that if
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 3then x=Dx
D, y=Dy
Dand z=Dz
Dwhere D=
∣
∣
∣
∣
∣a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣
∣
∣
∣
∣Dx=∣
∣
∣
∣
∣b 1 a 12 a 13
b 2 a 22 a 23
b 3 a 32 a 33∣
∣
∣
∣
∣i.e. thex-column has been replaced by the R.H.S.
bcolumn,
Dy=∣
∣
∣
∣
∣a 11 b 1 a 13
a 21 b 2 a 23
a 31 b 3 a 33∣
∣
∣
∣
∣i.e. they-column has been replaced by the R.H.S.
bcolumn,
Dz=∣
∣
∣
∣
∣a 11 a 12 b 1
a 21 a 22 b 2
a 31 a 32 b 3∣
∣
∣
∣
∣i.e. thez-column has been replaced by the R.H.S.
bcolumn.
Problem 7. Solve the following simultaneous
equations using Cramers rulex+y+z= 4
2 x− 3 y+ 4 z= 33
3 x− 2 y− 2 z= 2(This is the same as Problem 2 and a comparison
of methods may be made). Following the above
method:D=∣
∣
∣
∣
∣111
2 − 34
3 − 2 − 2∣
∣
∣
∣
∣=1(6−(−8))−1((−4)−12)
+1((−4)−(−9))= 14 + 16 + 5 = 35Dx=∣
∣
∣
∣
∣411
33 − 34
2 − 2 − 2∣
∣
∣
∣
∣=4(6−(−8))−1((−66)−8)
+1((−66)−(−6))= 56 + 74 − 60 = 70Dy=∣
∣
∣
∣
∣14 1
233 4
32 − 2∣
∣
∣
∣
∣=1((−66)−8)−4((−4)−12)+1(4−99)
=− 74 + 64 − 95 =− 105Dz=∣
∣
∣
∣
∣114
2 − 333
3 − 22∣
∣
∣
∣
∣=1((−6)−(−66))−1(4−99)
+4((−4)−(−9))= 60 + 95 + 20 = 175Hencex=Dx
D=70
35= 2 , y=Dy
D=− 105
35=− 3andz=Dz
D=175
35= 5Now try the following exercise.Exercise 115 Further problems on solving
simultaneous equations using Cramers rule- Repeat problems 3, 4, 5, 7 and 8 of Exercise
113 on page 279, using Cramers rule. - Repeat problems 3, 4, 8 and 9 of Exercise 114
on page 282, using Cramers rule.