How to find row echelon form of a matrix?

A matrix is said to be in a row-echelon form if it satisfies the following conditions:

• For a non-zero row each leading entry(first non-zero element) should have all zero elements to its left and bottom.

• All zero rowsRows in which all elements are 0. should be placed at the end of the matrix.

For a matrix A, we represent this as:

Here the leading entry for the 1st row and 2nd row is 1 and 2, respectively, and the elements below and to the left are 0. The zero row is placed at the end.

Basic and non-basic column

In a matrix of row echelon form, basic columns are columns that contain a leading entry, while non-basic columns are columns that don't have a leading entry.

In the following example of matrix $A$:

Here the 1st column is a basic column with leading entry$1$while 2nd column is a non-basic column with no leading entry.

Calculating the row-echelon form of a matrix

We use the Gaussian elimination method to calculate the row-echelon form of a matrix. This method mainly involves the following steps:

• Interchanging any two rows

• Adding any two rows

• Multiplying one row by a non-zero scalarA numerical value..

Note: We can perform any two of the above steps together at the same time.

Example

Let's consider an example to convert a matrix to a row-echelon form. Suppose we have a matrix$A$such that:

• The leading entry of the 1st row is 1, we convert all the elements below it to 0.

Adding the 1st row multiplied by -3 to the 2nd row:

Adding the 1st row multiplied by -1 to the 3rd row:

• The leading entry of the 2nd row is -1 and all the elements below it and on its left are 0, so our matrix is now in row echelon form.

Row echelon form of matrix A:

Conclusion

The row echelon form of a matrix simplifies calculations and helps us understand essential matrix properties. It's achieved by performing row operations to create a staircase-like pattern of leading coefficients and zeros. Moreover, the row echelon form finds application in various areas of mathematics to solve linear systems, determine rank, and understand matrix properties.