Good Systems Of Linear Equations - Cramer’s Rule Research Paper Example

Introduction

Cramer’s rule is the method for solving linear equations based on manipulation of determinants. The biggest advantage of using Cramer’s rule is that it provides solutions to individual variables independently of others. It also helps in finding the solutions of equations easily with irrational coefficients.
The Cramer’s rule states that the system of linear equations in matrix form Ax=b can be solved by finding the unknown variables using quotients of determinants. The numerator contains the determinant derived from A by replacing the ith unknown column with the vector b and the denominator contains the determinant of A itself.
The Cramer’s rule might seem wondrous on the first look, but it does carry some limitations. First, the system of linear equations must have same number of equations as unknowns to have square coefficient matrix. Secondly, the determinant of coefficient matrix must be non-zero. Otherwise, the solution cannot be determined from Cramer’s rule.
In this report, we will solve two systems of linear equations involving 4 unknown variables in each. We will find the determinants of 4×4 matrices and then use Cramer’s rule to find the values of unknown variables.

Method

Consider the system of linear equations:
3a-4b+c-d=1
-a+b-c+d=-2
2a+c+d=2
-3a-3b-c+d=-9

The coefficient matrix in this case is:

A=3 -4 1 -1-1 1 -1 12 0 1 1-3 -3 -1 2

The output vector is:

b=1-22-9

The determinant of coefficient matrix is 33.

The solutions to individual unknown variables are computed as follows:
a=1 -4 1 -1-2 1 -1 12 0 1 1-9 -3 -1 23 -4 1 -1-1 1 -1 12 0 1 1-3 -3 -1 2=1
b=3 1 1 -1-1-2-1 12 2 1 1-3 -9 -1 23 -4 1 -1-1 1 -1 12 0 1 1-3 -3 -1 2=1
c=3 -4 1 -1-1 1 -2 12 0 2 1-3 -3 -9 23 -4 1 -1-1 1 -1 12 0 1 1-3 -3 -1 2=1
d=3 -4 1 1-1 1 -1 -22 0 1 2-3 -3 -1 -93 -4 1 -1-1 1 -1 12 0 1 1-3 -3 -1 2=-1

Now consider the second system of linear equations:

3m-3n+l-f=-4
-m+2n-l+2f=8
2m+n-3l=-6
4m-5n-l+2f=-1

The coefficient matrix in this case is:

A=3-3-12 1-1-12214-5 -30-12

The output vector is:

b=-48-6-1

The determinant of coefficient matrix is -64.

The solutions to individual unknown variables are computed as follows:
m=-4-382 1-1-12-61-1-5 -30-123-3-12 1-1-12214-5 -30-12=0.8906
n=3-4-18 1-1-122-64-1 -30-123-3-12 1-1-12214-5 -30-12=1.9219
l=3-3-12 -4-182214-5 -60-123-3-12 1-1-12214-5 -30-12=3.2344
f=3-3-12 1-4-18214-5 -3-6-1-13-3-12 1-1-12214-5 -30-12=4.1406

Conclusions

It is clear from the application of Cramer’s rule to the system of linear equations that it is one of the easiest methods to find solutions of unknown variables independently from others even if they carry irrational values.

References

Bronson, R. (1991). Matrix methods: an introduction. Gulf Professional Publishing.
Young, C. Y. (2012). College Algebra. John Wiley & Sons.