Linear Algebra
Chapter 2: Row Echolon Form
Chapter 3: Reduced Row Echolon Form
Chapter 4: Rank of a Matrix
Chapter 5: Solution of System of Equations
Chapter 6: Eigenvalues and Eigenvactors
Other chapters will be updated soon.
Chapter 2: Row Echelon Form of a Matrix.
A matrix A is said to be in row echelon form if it satisfies the following properties:
i. Every zero row of the matrix A occurs below a non-zero row.
ii. The first non-zero number from the left of a non-zero row is a 1. This is called a leading 1.
iii. For each non-zero row, the leading 1 appears to the right and below any leading 1 in the preceding rows.
The below matrices are in echelon form.
Basic procedure to reduce a matrix in echelon form
Step
I: Make the first element 
of the matrix
unity.
Step
II: Make all the elements in first column below is zero.
Step III: Make
the element of second row and second column 
is unity. If is zero due to
previous row operation then make the next element of 2nd row unity.
Step IV: Make
all the element below unity 
is zero.
Step V: Follow the same procedure for next row.
Example 1: find the row echelon form of the matrix.
Solution: Given matrix is
Step
I: Make the first element of the matrix
unity.
As we can see the
third row has the first entry is 1. So interchange the row 1 and 3. i.e. 
Step II: Make all the elements in first column below is zero.
To make element 
we have the row
operation 
Since element 
is already zero
so we use row operation for element 
To make element 
we have the row
operation 
Step III: Make the element of second row and second column is unity.
To make 
we interchange
the rows 2 and three. Then multiply 
by -1 we get.
And
Step IV: Make all the element below unity is zero.
To make 
we have the operation 
Now interchanging the Row 3 and 4 and multiplying by 
, we get,
Now making 
i.e. 
The Row echelon form of the given matrix is
continued .........
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