System of Equations using Matrices

Consider the following system of in linear equations in n unknowns:

a11 x1 + a12 x2 + … + a1n xn = b1

a21 x1 + a22 x2 + … + a2n xn = b2

…        …        …        …   

…        …        …        …   

am1 x1 + am2 x2 + … + amn xn = bm

This system of equations can be written in matrix from as

Or, AX = B, where

And the m x n matrix A is called the coefficient matrix of the system of linear equations.

A set of values of the variables

x1, x2, …. xn which simultaneously satisfy all the equations is called a solution of the system of equations.

For example, x = 2, y = -3 is a solution of the system of linear equations

3x + y = 3, 2x + y = 1

Because 3(2) + (-3) = 3 and 2(2) + (-3) = 1

HOMOGENEOUS AND NON-HOMOGENEOUS SYSTEMS OF LINEAR EQUATIONS:

A system of equations AX=B is called a homogeneous system if B=0.

Otherwise, it is called a non-homogeneous system of equations.

For example, the system of equations

2x + 3y = 0, 3x — y = 0

Is a homogeneous system of linear equations whereas the system of equations given by

2x + 3y = 1, 3x — y = 5

Is a non-homogenous system of linear equations.

ALGORITHM

STEP I: Obtain the system of equations and express it in the matrix from AX=B.

STEP II: Find |A|.

STEP III: If |A| ≠ 0, then compute A-1 = 1/ |A| (adj A)  

STEP IV: Use the formula X = A-1B to obtain the unique solution.

EXAMPLE:

Determine the product

And use it to solve the system of equations:

x – y + z = 4, x — 2y — 2z = 9, 2x + y + 3z = 1

SOLUTION:

Let

Then, the given product CA is given by

The given system of equations can be written in matrix form as

Or, AX = B, where

The solution of this system of equations is given by

X = A-1B

X = 3, y = -2 and z = -1