The solution of simultaneous equations by matrices and determinants 245
Following the procedure:
(i) ( 9 +j 12 )I 1 −( 6 +j 8 )I 2 − 5 = 0
−( 6 +j 8 )I 1 +( 8 +j 3 )I 2 −( 2 +j 4 )= 0
(ii)
I 1
∣
∣
∣
∣
−( 6 +j 8 ) − 5
( 8 +j 3 ) −( 2 +j 4 )
∣
∣
∣
∣
=
−I 2
∣
∣
∣
∣
( 9 +j 12 ) − 5
−( 6 +j 8 ) −( 2 +j 4 )
∣
∣
∣
∣
=
1
∣
∣
∣
∣
( 9 +j 12 ) −( 6 +j 8 )
−( 6 +j 8 )( 8 +j 3 )
∣
∣
∣
∣
I 1
(− 20 +j 40 )+( 40 +j 15 )
=
−I 2
( 30 −j 60 )−( 30 +j 40 )
=
1
( 36 +j 123 )−(− 28 +j 96 )
I 1
20 +j 55
=
−I 2
−j 100
=
1
64 +j 27
HenceI 1 =
20 +j 55
64 +j 27
=
58. 52 ∠ 70. 02 ◦
69. 46 ∠ 22. 87 ◦
= 0. 84 ∠ 47. 15 ◦A
and I 2 =
100 ∠ 90 ◦
69. 46 ∠ 22. 87 ◦
= 1. 44 ∠ 67. 13 ◦A
(b) When solving simultaneous equations inthree
unknowns using determinants:
(i) Write the equations in the form
a 1 x+b 1 y+c 1 z+d 1 = 0
a 2 x+b 2 y+c 2 z+d 2 = 0
a 3 x+b 3 y+c 3 z+d 3 = 0
and then
(ii) the solution is given by
x
Dx
=
−y
Dy
=
z
Dz
=
− 1
D
whereDxis
∣ ∣ ∣ ∣ ∣ ∣
b 1 c 1 d 1
b 2 c 2 d 2
b 3 c 3 d 3
∣ ∣ ∣ ∣ ∣ ∣
i.e. the determinant of the coefficients
obtained by covering up thexcolumn.
Dyis
∣ ∣ ∣ ∣ ∣ ∣
a 1 c 1 d 1
a 2 c 2 d 2
a 3 c 3 d 3
∣ ∣ ∣ ∣ ∣ ∣
i.e., the determinant of the coefficients
obtained by covering up theycolumn.
Dzis
∣ ∣ ∣ ∣ ∣ ∣
a 1 b 1 d 1
a 2 b 2 d 2
a 3 b 3 d 3
∣ ∣ ∣ ∣ ∣ ∣
i.e. the determinant of the coefficients
obtained by covering up thezcolumn.
andDis
∣ ∣ ∣ ∣ ∣ ∣
a 1 b 1 c 1
a 2 b 2 c 2
a 3 b 3 c 3
∣ ∣ ∣ ∣ ∣ ∣
i.e. the determinant of the coefficients
obtained by covering up the constants
column.
Problem 6. A d.c. circuit comprises three closed
loops. Applying Kirchhoff’s laws to the closed
loops gives the following equations for current flow
in milliamperes:
2 I 1 + 3 I 2 − 4 I 3 = 26
I 1 − 5 I 2 − 3 I 3 =− 87
− 7 I 1 + 2 I 2 + 6 I 3 = 12
Use determinants to solve forI 1 ,I 2 andI 3.
(i) Writing the equations in the
a 1 x+b 1 y+c 1 z+d 1 =0formgives:
2 I 1 + 3 I 2 − 4 I 3 − 26 = 0
I 1 − 5 I 2 − 3 I 3 + 87 = 0
− 7 I 1 + 2 I 2 + 6 I 3 − 12 = 0
(ii) the solution is given by
I 1
DI 1
=
−I 2
DI 2
=
I 3
DI 3
=
− 1
D
where DI 1 is the determinant of coefficients
obtained by covering up theI 1 column, i.e.