Cramer’s Rule
A.1 Cramer’s Rule
In many cases, Cramer’s rule can be used to solve the simultaneous equa- tions we encounter in circuit analysis. Cramer’s rule states that the solution to Eq. (A.1) or (A.3) is
x 2 ⫽¢ 2 (A.5)
x n ⫽¢ n ¢
Appendix A Simultaneous Equations and Matrix Inversion
A-1
where the ’s are the determinants given by ¢
Notice that ¢ is the determinant of matrix A and ¢ k is the determi- nant of the matrix formed by replacing the kth column of A by B. It is evident from Eq. (A.5) that Cramer’s rule applies only when ¢ ⫽ 0. When ¢⫽0 , the set of equations has no unique solution, because the equations are linearly dependent.
The value of the determinant , for example, can be obtained by ¢ expanding along the first row:
a 11 a 12 a 13 pa 1n
a 21 a 22 a 23 pa 2n
¢⫽5 a 31 a 32 a 33 pa 3n 5 (A.7)
oo o p o
a n 1 a n 2 a n 3 pa nn ⫽a 11 M 11 ⫺a 12 M 12 ⫹a 1⫹n 13 M 13 ⫹ p ⫹ (⫺1) a 1n M 1n
where the minor M ij is an (n ⫺ 1) ⫻ (n ⫺ 1) determinant of the matrix formed by striking out the ith row and jth column. The value of ¢ may also be obtained by expanding along the first column:
n ¢⫽a ⫹1 11 M 11 ⫺a 21 M 21 ⫹a 31 M 31 ⫹ p ⫹ (⫺1) a n 1 M n 1 (A.8)
We now specifically develop the formulas for calculating the deter- minants of 2⫻2 and 3⫻3 matrices, because of their frequent occur- rence in this text. For a 2⫻2 matrix,
¢⫽2 2⫽a 11 a 22 ⫺a 12 a 21 a (A.9)
a 11 a 12
21 a 22
For a 3⫻3 matrix,
a 11 a 12 a 13
¢⫽3 a 21 a 22 a 23 3⫽a 11 (⫺1) 2 2⫹a 21 (⫺1) 2 a 2
2 a 22 a 23 3 a 12 a 13
32 a 33 a 32 a 33
a 31 a 32 a 33
a 12 a 13
⫹a 31 (⫺1) 4 2 2
a 22 a 23 ⫽a 11 (a 22 a 33 ⫺a 32 a 23 )⫺a 21 (a 12 a 33 ⫺a 32 a 13 ) ⫹a 31 (a 12 a 23 ⫺a 22 a 13 )
A-2
Appendix A Simultaneous Equations and Matrix Inversion
An alternative method of obtaining the determinant of a 3⫻3 matrix is by repeating the first two rows and multiplying the terms diagonally as follows.
⫺a 33 a 12 a 21 (A.11)
In summary:
The solution of linear simultaneous equations by Cramer’s rule boils down to finding
x k ⫽ ¢ k , k ⫽ 1, 2, . . . , n ¢ (A.12)
where ¢ is the determinant of matrix
A and ¢ k is the determinant of the matrix formed by replacing the k th column of
A by B.
One may use other methods, such You may not find much need to use Cramer’s method described in as matrix inversion and elimination.
this appendix, in view of the availability of calculators, computers, and Only Cramer’s method is covered
software packages such as MATLAB, which can be used easily to solve here, because of its simplicity and
a set of linear equations. But in case you need to solve the equations also because of the availability of
by hand, the material covered in this appendix becomes useful. At any powerful calculators.
rate, it is important to know the mathematical basis of those calcula- tors and software packages.
Example A.1
Solve the simultaneous equations
The given set of equations is cast in matrix form as
4 ⫺3 x 1 c 17 ⫺ 3 d c 5 x 2 d⫽c ⫺ 21 d
The determinants are evaluated as
Appendix A Simultaneous Equations and Matrix Inversion
Find the solution to the following simultaneous equations:
Practice Problem A.1
3x 1 ⫺x 2 ⫽
4, ⫺6x 1 ⫹ 18x 2 ⫽ 16
Answer: x 1 ⫽ 1.833, x 2 ⫽ 1.5.
Determine x 1 ,x 2 , and x 3 for this set of simultaneous equations:
Example A.2
25x 1 ⫺ 5x 2 ⫺ 20x 3 ⫽ 50 ⫺5x 1 ⫹ 10x 2 ⫺ 4x 3 ⫽ 0 ⫺5x 1 ⫺ 4x 2 ⫹ 9x 3 ⫽0
Solution:
In matrix form, the given set of equations becomes
25 ⫺5 ⫺20 x 1 50 £ ⫺ 5 10 ⫺ 4 § £ x 2 §⫽£ 0 §
We apply Eq. (A.11) to find the determinants. This requires that we repeat the first two rows of the matrix. Thus,
⫽ 2250 ⫺ 400 ⫺ 100 ⫺ 1000 ⫺ 400 ⫺ 225 ⫽ 125 Similarly,
A-4
Appendix A Simultaneous Equations and Matrix Inversion
Hence, we now find
Practice Problem A.2
Obtain the solution of this set of simultaneous equations: 3x 1 ⫺x 2 ⫺ 2x 3 ⫽ 1
⫺x 1 ⫹ 6x 2 ⫺ 3x 3 ⫽ 0 ⫺2x 1 ⫺ 3x 2 ⫹ 6x 3 ⫽6
Answer: x 1 ⫽ 3⫽x 3 ,x 2 ⫽ 2 .