{"id":4612,"date":"2021-08-03T10:13:07","date_gmt":"2021-08-03T04:43:07","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=4612"},"modified":"2026-03-23T05:09:23","modified_gmt":"2026-03-22T23:39:23","slug":"what-is-the-echelon-form-of-a-matrix","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-echelon-form-of-a-matrix\/","title":{"rendered":"What is the echelon form of a matrix?"},"content":{"rendered":"\n

Overview<\/strong>(echelon form)<\/h2>\n\n\n\n

There are two types of echelon form of a matrix<\/a>.<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Row echelon form<\/strong><\/p>\n\n\n\n

A matrix satisfying the following conditions is said to be in the row echelon form-<\/p>\n\n\n\n

Condition-1:  The first non-zero element (leading element) in each row should be 1.<\/p>\n\n\n\n

Condition-2: Each leading element is in a column to the right of the leading element in the previous row.<\/p>\n\n\n\n

Condition-3: The rows with all zero elements (if any) are at the bottom.<\/p>\n\n\n\n

Inverse of a matrix by using elementary transformation<\/strong><\/h2>\n\n\n\n

The following transformation, we call as elementary transformations-<\/p>\n\n\n\n

1. Interchange of any two rows (column)<\/p>\n\n\n\n

2. Multiplication of any row or column by any non-zero scalar quantity k.<\/p>\n\n\n\n

3. Addition to one row (column) of another row(column) multiplied by any non-zero scalar.<\/p>\n\n\n\n

The symbol ~ is used for equivalence.<\/p>\n\n\n\n

Elementary matrices<\/strong><\/p>\n\n\n\n

If we get a square matrix from an identity or unit matrix by using any single elementary transformation is called elementary matrix.<\/p>\n\n\n\n

Note- Every elementary row transformation of a matrix can be affected by pre multiplication with the corresponding elementary matrix.<\/p>\n\n\n\n

Rank of a matrix by echelon form<\/strong><\/p>\n\n\n\n

The rank of a matrix (r) is, when\u2013<\/p>\n\n\n\n

1. It has at least one non-zero minor of order r.<\/p>\n\n\n\n

2. Every minor of A of order higher than r is zero.<\/p>\n\n\n\n

Solved examples<\/h2>\n\n\n\n

Example: Find the rank of a matrix M<\/strong>.<\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Sol. First we will convert the matrix M into echelon form,<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

We can see that, in this echelon form of matrix, the number of non \u2013 zero rows is 3.<\/p>\n\n\n\n

So that the rank of matrix X will be 3.<\/p>\n\n\n\n

Example: Find the rank of the following matrices by echelon form?<\/strong><\/p>\n\n\n

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\"\"\/<\/figure><\/div>\n\n\n

Sol<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

It is  clear  that  minor  of order 3  vanishes but  minor  of order  2  exists  as  <\/p>\n\n\n

\n
\"\"\/<\/figure><\/div>\n\n\n

Hence rank of a given matrix A is 2.<\/p>\n\n\n\n

Int<\/strong>erested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n