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									    LINEAR ALGEBRA
         &
DIFFERENTIAL EQUATIONS

        SPRING – 2012
             Lecture -3
           (Second Week)


  For any clarification please contact:

         Zain Ul abadin Zafar
Elementary Row Operations:
                       The following operations on a matrix “A”
are called elementary row operations:
  Interchange of any two rows of “A”
  Multiplication of a row by a non-zero real or complex
  number called scalar
  Addition of a scalar multiples of one row of “A” to
  another row
Let “A” be an mxn matrix. An mxn matrix “B” is called row
eqivalent to “A” if “B” is obtained from “A” by performing a
finite sequence of elementary operations on “A”. We denote
“B” is equivalent to “A” as:
e.g
     Read    is row equivalent to .
Echelon Form:
             A matrix is said to be in echelon form if:
     The first non-zero entry in each row is 1
     If row “k” does not contain entirely zeros, the
     number of leading zeros entries in row k+1 is greater
     than the number of leading zeros entries in row k
     If there are rows whose entries are all zero, they are
     below the rows having non-zero entries
e.g.




Reduced Echelon Form:
                      A matrix is said to be in reduced
echelon form if:
   The matrix is in row echelon form
   The first non-zero entry(leading entry) in each
   row is the only non-zero entry in its column
   e.g.
Example-1



Reduce the matrix A=   to echelon form.



Solution:
EXAMPLE-2
Reduce the following matrix into reduced echelon form:
Solution:

       1.
Which is required reduced echelon form.

								
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