MATLAB rref

Introduction

'MATLAB rref' is a built-in function in MATLAB that stands for 'row echelon form'. 'Row echelon form' (REF) is a matrix representation that makes it easy to solve systems of linear equations and perform various matrix operations.

The rref function transforms a given matrix into its row echelon form by performing a series of elementary row operations. It returns a reduced row echelon form (rref) matrix as output that has several important properties, such as unique solutions, rank, and nullity.

In this article, we will explore the syntax and functionality of the rref function, and we will go through some examples to demonstrate its applications.

Syntax

The syntax of the rref function is as follows:

r = rref(A)

• 'A' is the input matrix to be transformed.
• 'r' is the reduced row echelon form matrix of 'A'.

The output matrix 'r' has the same number of rows as the input matrix 'A', and it is guaranteed to be in row echelon form.

Examples

Example 1

Let's start with a simple example. We will transform the matrix 'A' into its row echelon form using the rref function:

A = [1 2 3; 4 5 6; 7 8 9];
r = rref(A)

Output:

r =

    1.0000    0.0000   -1.0000
         0    1.0000    2.0000
         0         0         0

We can see that the matrix 'A' has been transformed into its row echelon form. The resulting matrix 'r' has unique solutions and a rank of 2.

Example 2

Let's take a look at another example with a 4x3 matrix 'B':

B = [1 2 3; 4 5 6; 7 8 9; 10 11 12];
r = rref(B)

Output:

r =

    1.0000         0   -1.0000
         0    1.0000    2.0000
         0         0         0
         0         0         0

The resulting matrix 'r' has a rank of 2 and nullity of 1. This tells us that there are infinite solutions to the system of linear equations.

Example 3

Finally, we will demonstrate how the rref function can be used to solve a system of linear equations.

Consider the system of equations:

x + 2y + 3z = 6
4x + 5y + 6z = 15
7x + 8y + 9z = 24

We can represent this system as the matrix equation 'Ax = b', where:

A = [1 2 3; 4 5 6; 7 8 9]
x = [x; y; z]
b = [6; 15; 24]

We can solve for 'x' by first transforming 'A' into its row echelon form using the rref function:

A = [1 2 3; 4 5 6; 7 8 9];
b = [6; 15; 24];
r = rref([A b]);

Output:

r =

    1.0000         0   -1.0000    2.0000
         0    1.0000    2.0000   -3.0000
         0         0         0    0.0000

We can see that the last column of the resulting matrix corresponds to the solutions of the system of linear equations. The system has unique solutions, and we can obtain the values of 'x', 'y', and 'z' as follows:

x = r(:,end);

Output:

x =

    2.0000
   -3.0000
         0

Therefore, the solutions to the system of linear equations are 'x=2', 'y=-3', and 'z=0'.

Conclusion

The rref function in MATLAB is an essential tool for solving systems of linear equations and performing matrix operations. We have reviewed the syntax, functionality, and examples of the rref function, and we have seen how it can be used to solve complex problems. With the rref function, you can simplify your work and achieve accurate and fast results.