Linear Algebra Chapter 3 Reduced Row Echelon Form

 

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Linear Algebra                                       

Chapter 1: Matrix

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 3: Reduced Row Echelon Form of a Matrix.

Before study of this chapter study RowEchelon Form of a Matrix.

A matrix A is said to be in Reduced Row Echelon form if it satisfies the following properties addition to properties of Row Echelon Form.

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.

iv.            Every element of the column having leading element must be zero. 

The below matrices are in echelon form.

Basic procedure to reduce a matrix in Reduced Row echelon form

Steps: follow the steps I to V mentioned in formation of Row Echelon Form   

In addition to mentioned steps make each element zero in column having leading element 1.  

Example 1: Find the Reduced row echelon form of the matrix.

Solution: The Row echelon form the given matrix (givenin Chapter 2) is

Now making all the entries in column zero except leading element by row operation.

We get the last column entry zero form of the matrix by following Row operation.

And                                                 

Now by using following row operations we get the third column zero except leading element.

Finally, the last element in column 2 becomes zero by , we get.