146 Nodal and mesh analysis
2Ii - 41 4i 1
Solution
This is a square matrix of order 3. The elements in column I (n = 1) are all
a21 - 0; a31 = 4. From Equation (7.1), taking n - 1,
3
A -- m=l ~, amlCml -- allCll q- a21c21 -Jr- a31c31
Now Cll is the cofactor of element a~l and by definition
= 2;
Cl,- (-1) C1+1) a22 a23
a32 a33
Similarly
C21 -- (__1)(2+1) a12 a13
a32 a33
and c31- (--1) (3+1)
a12 a13
a22 a23
Therefore
A- 2(-1) 2
4 -2
-1 -1
+ (0 X c21 ) + 4(_1)41-21
4
9 I 4 -2
= (2)[-4- 2] + (4)[4- 16]
= -12 - 48 - -60
Cramer's rule
This is most useful for solving simultaneous equations using matrix methods. A
set of simultaneous equations with unknowns in x may be written
Ax- B
where A is a rectangular matrix of the coefficients of x and B is a column
matrix. Cramer's rule states that to find Xm we obtain the determinant A of A
and divide it into A m, where A m is the determinant of the matrix obtained by
replacing the mth column of A by the column matrix B.
Example 7.8
The three mesh equations of a certain circuit are
51~+ 212+ 13- 5
I~ + 1012-213- 10
21~- 312+41~- 0
Determine the current I~.