Overview(echelon form)
There are two types of echelon form of a matrix.
Row echelon form
A matrix satisfying the following conditions is said to be in the row echelon form-
Condition-1: The first non-zero element (leading element) in each row should be 1.
Condition-2: Each leading element is in a column to the right of the leading element in the previous row.
Condition-3: The rows with all zero elements (if any) are at the bottom.
Inverse of a matrix by using elementary transformation
The following transformation, we call as elementary transformations-
1. Interchange of any two rows (column)
2. Multiplication of any row or column by any non-zero scalar quantity k.
3. Addition to one row (column) of another row(column) multiplied by any non-zero scalar.
The symbol ~ is used for equivalence.
Elementary matrices
If we get a square matrix from an identity or unit matrix by using any single elementary transformation is called elementary matrix.
Note- Every elementary row transformation of a matrix can be affected by pre multiplication with the corresponding elementary matrix.
Rank of a matrix by echelon form
The rank of a matrix (r) is, when–
1. It has at least one non-zero minor of order r.
2. Every minor of A of order higher than r is zero.
Solved examples
Example: Find the rank of a matrix M.
Sol. First we will convert the matrix M into echelon form,
We can see that, in this echelon form of matrix, the number of non – zero rows is 3.
So that the rank of matrix X will be 3.
Example: Find the rank of the following matrices by echelon form?
Sol
It is clear that minor of order 3 vanishes but minor of order 2 exists as
Hence rank of a given matrix A is 2.
Interested in learning about similar topics? Here are a few hand-picked blogs for you!