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Chapter 3

System of Linear Equations and

Matrices

3.1 Introduction to vectors and matrices

3.2 Existence and Uniqueness of Solutions

• Matrix algebra is used to solve a system of simultaneous linear

equations. A general set of m linear equations and n unknowns,

A system of equations can be consistent or inconsistent. What does that

mean?

• A system of equations [ A ][ X ]=[ C ] is consistent if there is a solution, and it is inconsistent if there is no solution. However, a consistent system of equations does not mean a unique solution, that is, a consistent system of equations may have a unique solution or infinite solutions. How can one distinguish between a consistent and inconsistent system of equations?
• A system of equations [ A ][X]=[ C ] is consistent if the rank of A is equal to the rank of the augmented matrix [ A:C ].
• A system of equations [ A ][X]=[ C ] is inconsistent if the rank of A is less than the rank of the augmented matrix [ A:C ].
• But, what do you mean by rank of a matrix? The rank of a matrix is defined as the order of the largest square sub matrix whose determinant is not zero.

• Example 1: What is the rank of

• Hence the rank of the augmented matrix [ B ] is 3. Since [ A ] = [ D ] , the rank of [ A ] is 3. Since the rank of the augmented matrix [ B ] equals the rank of the coefficient matrix [A], the system of equations is consistent.

• Example 3: Use the concept of rank to find if the equation is consistent or inconsistent.
• Solution The augmented matrix is
• Since there are no square sub matrices of order 4 x 4 as the augmented matrix [B] is a 4 x 3 matrix, the rank of the augmented matrix [ B ] is at most 3. So let us look at square sub matrices of the augmented matrix (B) of order 3 and see if any of the 3 x 3 sub matrices have a determinant not equal to zero.

• So the rank of the coefficient matrix [ A ] is 2.
• Since the rank of the coefficient matrix [ A ] is less than the rank of the augmented matrix [B] , the system of equations is inconsistent. Hence, no solution exists for [A ][ X ] = [ C ].

If a solution exists, how do we know whether it is unique?

• Consistent systems of equations can only have a unique solution or infinite solutions. Can a system of equations have more than one but not infinite number of solutions? No, you can only have either a unique solution or infinite solutions.

3.3 Methods of solution of Linear Equations

3.3.1 Gaussian Elimination

How is a set of equations solved numerically?

• One of the most popular techniques for solving simultaneous linear equations is the Gaussian elimination method. The approach is designed to solve a general set of n equations and n unknowns.

• This is the end of the first step of forward elimination. Now for the second step of forward elimination, we start with the second equation as the pivot equation and a ’ 22 as the pivot