Matrix Algebra to solve systems of linear equations

Matrix Algebra can be used to solve a system of linear equations. Matrix is a row and column of numbers and can be defined as n*m matrix where n is the number of rows and m is the number of columns.

In Matrix Algebra if A is a matrix which is a square matrix it may have an inverse depending on the values of rows and columns. If a matrix do not have an inverse then the system of equations do not have a real number solution. However, they may have complex number solution.

That is in Matrix Algebra to solve a system of equations one has to first evaluate whether it has an inverse. If it has an inverse then one must have a method to find out the inverse to find the solution to the problem.

The Gauss/Jordan method is used to find the solution to a system of linear equations. Matrix Algebra is useful in Economics, Statistics, and Engineering and in Science. This is because the matrix can be easily used in a computer program and if one uses algorithms then it can get solutions faster irrespective of the matrix dimensions. However, if the matrix is very large even a computer may have difficulty in finding an inverse due to the complex manipulations and steps to get the solutions.

Gauss/Jordan Method of finding solution to a system of linear equation

First step in finding the solution is to determine whether the matrix or system of linear equations has an inverse. That is first to find the determinant of a matrix, which is a scalar value not a vector. If a matrix does not have a zero value, then that matrix has an inverse. However, to find the determinant is not easy if that square matrix is very large.

The second step is to convert the system of equations in to an augmented matrix by entering the coefficients of the variables as vectors and use the constants in an augmented matrix form.

Then start pivoting the columns where there is 1 in the diagonal of the matrix until one gets the identity matrix. One can see from the process of matrix manipulation method of the Gaussian transformation process of the matrix, a matrix can be manipulated using an algorithm and can be converted in to a computer program. This is one of the advantages of matrix method to find solutions to a system of linear equations. However, if the system of equations or the matrix is very large then even a super computer has technical and mathematical problems of generating solutions in a reasonable time frame. This is caused because of finding a determinant is a time consuming complex process due to the number of operations gets exponentially large if the rows and columns is increasing.

One can see from the above the feasibility in solving simultaneous equations depends on the number of variables to be evaluated and the number of linear equations. However, in modern times because of the speed of the computers very large matrix manipulations are possible. Even now, given the computer technological advances even a super computer has difficulty in finding solutions or finding the determinant of a matrix, if the matrix is very large compared to the capacity and the speed of processing the data by modern computers or super computers.

Example of Gauss/Jordan method of finding solution to a system of linear equations

Say in a problem one has to find three variable values given there relationships is three different equations as follows

x- y + 2z = 5

3x +2y +z = 10

2x-3y-2z = -10

 First step is to convert this matrix in to an augmented matrix form as below

[1 -1 2 | 5]    In this matrix see the first row of the first column it has a value of 1. This

[3 2 5 | 10]    is the first pivoting or manipulation to bring in the first column all zeros by

[2 -3-2|-10] doing appropriate arithmetic operation and adding or subtracting the rows

In this instance r2 – 3r1 = R2 and r3 – 2r1 = R3 which will give zeros in the first column.

After this process the matrix becomes as follows.

[1 -1 2| 5]          See in the second row of the second column is the diagonal. To transform

[0 5 -5| -5]         in to a identity matrix the value of 5 must be manipulated to 1 by

[0 -1 -6| -20]     diving it by 5. That is, 1/5 r2 =R2. After this pivoting the matrix becomes as follows.

[1 -1 2| 5]           See in this matrix in the second column other than the diagonal other

[0 1 -1| -1]          vectors are non-zeros. This can be made zero by r1+ r2 = R1 and

[0 -1 -6| -20]       r3 + r2 = R3. After manipulation the matrix will be as follows.

[1 0 1| 4]             See the 3rd column. It has a vector value of -7. This is in the diagonal.

[0 1 -1| -1]          This has to be transformed in to 1. This can be done by -1/7r3 = R3.

[0 0 -7| -21]        After this process this matrix becomes as follows.

[1 0 1| 4]             See in the third column other than the diagonal values are not zero.

[0 1 -1| -1]           Other values in the diagonals must be transformed to zeros.

[0 0 1| 3]              This can be done by r1 – r3 = R1 and r2+ r3 = R2.

After transformation the matrix becomes as follows

[1 0 0 | 1]           See the first three columns and the three rows. The diagonal values are 1.

[0 1 0|  2]           other values are zero. That is the original matrix has become an identity

[0 0 1| 3]            matrix. There fore the x= 1 y=2 and z = 3.

One can see from the above example even with three linear systems of equations the number of steps and manipulations are quite large compared to the simplicity of the matrix. Imagine you have 10 linear systems of equations then the number of arithmetic operations become quite large compared to the example given above. However, if it can be converted in to an algorithm and if it can be manipulated by computer processes, then a very large system of linear equations can be solved.  In addition, as mentioned above to

Evaluate the determinant of a matrix is not easy when the matrix becomes quite large.

Conclusion

As explained above, matrix algebra is a useful tool in solving mathematical problems. It is also useful in statistical analysis and also in finding solutions to scientific linear equations if it has some unknown variable. The matrix also enables to find solutions by creating algorithms, which is capable to be transformed in to a computer program or sub routine. Gauss/ Jordan method as explained above demonstrates the process of finding the solution by converting the augmented matrix in to an identity matrix if possible to find the solutions. One has to note all square matrix cannot be converted in to identity matrix if it does not have an inverse matrix. If any augmented matrix cannot be converted in to an identity matrix then the system of linear equations does not have a real solution.