Using the row elementary operations, we can transform a given non-zero matrix to a simplified form called a Row-echelon form.

**Row-Echelon form**

Using the row elementary operations, we can transform a given
non-zero matrix to a simplified form called a **Row-echelon form**. In a row-echelon
form, we may have rows all of whose entries are zero. Such rows are called **zero rows**. A non-zero row is
one in which at least one of the entries is not zero. For instance, in the
matrix,

,

*R*_{1} and *R*_{2} are non-zero rows and *R*_{3}
is a zero row

A non-zero matrix *E
*is said to be in a **row-echelon form **if:

i.
All zero rows of *E *occur below every non-zero row of *E*.

ii.
The first non-zero element in any row *i *of *E *occurs
in the *j*^{th} column of *E *, then all other entries in the
*j*^{th} column of *E *below the first non-zero element of
row *i *are zeros.

iii.
The first non-zero entry in the *i*th row of *E *lies
to the left of the first non-zero entry in (*i *+1)^{th} row of *E *.

**Note**

**A non-zero matrix is in a row-echelon form if all zero rows
occur as bottom rows of the matrix, and if the first non-zero element in any
lower row occurs to the right of the first non- zero entry in the higher row.**

The following matrices are in row-echelon form:

Consider the matrix in (i). Go up row by row from the last row of
the matrix. The third row is a zero row. The first non-zero entry in the second
row occurs in the third column and it lies to the right of the first non-zero
entry in the first row which occurs in the second column. So the matrix is in
row- echelon form.

Consider the matrix in (ii). Go up row by row from the last row of
the matrix. All the rows are non-zero rows. The first non-zero entry in the
third row occurs in the fourth column and it occurs to the right of the first
non-zero entry in the second row which occurs in the third column. The first
non-zero entry in the second row occurs in the third column and it occurs to
the right of the first non-zero entry in the first row which occurs in the
first column. So the matrix is in row-echelon form.

The following matrices are not in row-echelon form:

Consider the matrix in (i). In this matrix, the first non-zero
entry in the third row occurs in the second column and it is on the left of the
first non-zero entry in the second row which occurs in the third column. So the
matrix is not in row-echelon form.

Consider the matrix in (ii). In this matrix, the first non-zero
entry in the second row occurs in the first column and it is on the left of the
first non-zero entry in the first row which occurs in the second column. So the
matrix is not in row-echelon form.

**Step 1**

Inspect the first row. If the first row is a zero row, then the
row is interchanged with a non-zero row below the first row. If *a*_{11}
is not equal to 0, then go to step 2. Otherwise, interchange the first row *R*_{1}
with any other row below the first row which has a non-zero element in the
first column; if no row below the first row has non-zero entry in the first
column, then consider *a*_{12} . If *a*_{12} is not
equal to 0, then go to step 2. Otherwise, interchange the first row *R*_{1}
with any other row below the first row which has a non-zero element in the
second column; if no row below the first row has non-zero entry in the second
column, then consider *a*_{13}. Proceed in the same way till we
get a non-zero entry in the first row. This is called **pivoting **and the first non-zero
element in the first row is called the **pivot **of the first row.

**Step 2**

Use the first row and elementary row operations to transform all
elements under the pivot to become zeros.

**Step 3**

Consider the next row as first row and perform steps 1 and 2 with
the rows below this row only.

Repeat the step until all rows are exhausted.

Reduce the matrix to a row-echelon form.

Note

This
is also a row-echelon form of the given matrix.

So,
a row-echelon form of a matrix is not necessarily
unique.

**Example 1.14**

Reduce the matrix to a row-echelon form.

Solution

Tags : Definition, Theorem, Formulas, Solved Example Problems | Elementary Transformations of a Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Row Echelon form | Definition, Theorem, Formulas, Solved Example Problems | Elementary Transformations of a Matrix

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