Solving simultaneos Equations

Introduction

In Algebra, Simultaneous equations are an important technique to solve problems which can be Mathematically formulated. Solving simultaneous equations in linear form if there is less equations is easy. However, if there are more unknown variables and the number of equations increase it becomes harder and time consuming. In this case matrix algebra is a useful technique as it can be converted in to an algorithm and in a programme. However, one must know how matrix algebra works to solve the problem or how to solve simple simultaneous equations in linear form using simple algebraic mehtods if the number of equations are few.

Simple Algebriac technique to solve simultaneous equations

Say there are two variables (x) and (y) where  3x + 4y = 25 and 4x - 2y = 13 and one wants to calculate the value or values of x and y. The first step is to eliminate (x) or (y)  by multiplying the first by the co-efficent of the second and the second by the first. Then add or subtract to have the equation only in terms of (x) or (y) variable to calculate (x) or (y) first. Then substituting the value of (x) or (y) in one of the simple equation to find the other value. This is easy for linear simultaneous equations which oonly have two variables. However, if the number of simpultaneous equations increase the the number of manipulations or steps to solve the problem increases faster. As well, it is difficult to computerize the process. In order to speed up the process one uses matrices as they are easy to be converted in to an algorithm and can be programeed. That is knowledge of matrix algebra is important to solve complex simultaneous equations in the linear form.

Basic Ideas of Matrix Algebra

A matrix contains rows and columns. Matrices can be square matrix or non-square matrix. Square matrix have the same rows and columns. Most linear equations can be converted in to a matrix form in a multiplication form. For example , say 2x + 3y + 5 z = 12 and x - 4y + 2 z = -8 and 2x - 3y - z = -12.

These equations can be expressed in Matrix form as follows:

( 2 3 5)  (x) =   (12)

(1 -4 2)  (y) =    (-8)

( 2 -3 -1)(z) =  (-12)

Say the first matrix is (A), the seconf matrix (B) and the third matrix is (C)

Multiply the two sides by the inverse of (A) denoted by (A) to the power -1  we get

(A)* Inverse (A) * (B) = (C)* inverse (A)

But according to Matrix algebra multiplication rule (A)* inverse (A) = I (identity matrix)

There fore I* B = (C)*inverse (A)

But I*B = B = B * I according to matrix Algebra.

There fore B = (C)* inverse (A)

That is if we know the inverse of (A) and multiply the inverse the constant matrix we get the unknown values of (x) (y) and (z) if (A) has an Inverse. If it does not have an Inverse then these equations do not have a solution or cannot be solved or they dont have a real number solution.

There fore if one knows a technique to find an inverse of a matrix then this can be converted in to an algorithm and can be converted in to a computer program and this can apply the algorithm and produce an inverse. Then multiplication of the constant matrix by the inverse will give the answer.

Inverse of a Matrix

One can use Gauss-Jordan elimination to get  the inverse of a matrix. This technique transform [ A/ I] by row operation ctransform it in to [I/Inverse (A).

Example

Find the inverse of ( 1 2)

                            (3 4)

Solution

[ 1 2 / 1 0]  Multiply the fist row by 3 and transform Row 2 by substracting row 1 *3

[ 3 4 / 0 1]  this will transform row 2 and give the following matrix ( R2 - 3* R1)

[ 1 2 / 1 0]

[ 0-2/ -3 1]  Multiply row to by (-1/2) to make -2 to 1, this will give the following matrix

[ 1 2 /  1      0]  Row 1 - 2 * row 2 will give the following matrix

[ 0 1 /3/2 -1/2]

[ 1 0 / -2      1]

[ 0 1 / 3/2 -1/2]

There fore the inverse of ( 1 2)   is ( -2    1 )   These two matrix will give the identity

                                    ( 3 4)      (3/2 -1/2)  matrix.

The same method can be applied to any square matrix in any number of rows and columns as one can see the preocess is to make by row operations to make the original matrix in to an identity matrix by comparing the rows and making row 1 coumn1 1 row 2 coumn 2 1 and so and others in the coloumn zeros.

However, all square matrix will not have an inverse. There fore first one must test whether it has an inverse. This can be done to calculate the determinant of a matrix. If the determinant is not zero the matrix will have an inverse. Otherwise it will not have an inverse. There fore one must have a method to determinant of a square matrix.