Elementary row operations

by Marco Taboga, PhD

Elementary row operations are used to transform a system of linear equations into a new system that has the same solutions as the original one (i.e., into an equivalent system).

There are three elementary operations:

• multiplying an equation by a non-zero constant;

• adding a multiple of an equation to another equation;

• interchanging two equations.

Notation

A system of linear equations in unknowns is written in matrix form aswhere:

• is the matrix of coefficients;

• is the vector of unknowns;

• is the vector of constants.

The rows of the system are the equationswhere:

• is the -th row of (it contains the coefficients of the -th equation);

• is the -th entry of .

Multiplying an equation by a non-zero constant

The first elementary operation we consider is the multiplication of an equation by a constant .

If the -th equation is the one being multiplied, then we substitute the equationwith the equation

The original matrix of coefficients and vector of constantsbecomeso that the new system is

The same result can be achieved as follows:

1. take the identity matrix ;

2. multiply the -th row of by and denote the transformed matrix thus obtained by :

3. pre-multiply both sides of the matrix form of the system by :

It can be easily verified that

In the lecture on Equivalent systems, we have proved that if is invertible, then the new system is equivalent to the original one.

But the matrix above is invertible (full-rank) because its rows are linearly independent (none of them can be written as a linear combination of the others).

Thus, multiplying an equation by a non-zero constant gives an equivalent system.

Adding a multiple of one equation to another equation

The second elementary row operation we consider is the addition of a multiple of an equation to another equation.

Suppose we want to add times the -th equation to the -th equation. Then we substitute the equationwith the equation

The original matrix of coefficients and vector of constants becomeso that the new system is

The same result can be achieved as follows:

1. take the identity matrix ;

2. add times the -th row of to the -th row of , and denote the transformed matrix thus obtained by :

3. pre-multiply both sides of the matrix equation by :

As before, we have thatand the new system is equivalent to the original one because is invertible (none of its rows can be written as a linear combination of the others).

In other words, we obtain an equivalent system by adding a multiple of one row to another row.

Interchanging two equations

The third elementary row operation we consider is the interchange of two equations.

We switch the -th equation with the -th equation

The original matrix of coefficients and vector of constantsbecomeso that the new system is

The same result can be obtained as follows:

1. take the identity matrix ;

2. switch the -th row of with the -th row ( in the original matrix), and denote the new matrix by :

3. pre-multiply both sides of the system by :

As for the previous elementary operations, we have thatand the new system is equivalent to the original one because is invertible (the rows of are the same of , but in a different order; they form the standard basis of the space of vectors).

To sum up, we obtain an equivalent system by interchanging two rows (two equations) of the system.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Suppose that is a system of equations in unknowns.

What is the matrix that allows us to interchange the second equation with the fourth (when the system is pre-multiplied by )?

Exercise 2

Suppose that we have a system of equations in unknowns.

What is the matrix that allows us to multiply the second equation by ?

Exercise 3

Suppose that we have a system of equations in unknowns.

What is the matrix that allows us to add the first equation to the second?

How to cite

Please cite as:

Taboga, Marco (2021). "Elementary row operations", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/elementary-row-operations.