1.3 The Gauss{Jordan algorithm

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We now describe the GAUSS{JORDAN ALGORITHM. This is a process

which starts with a given matrix A and produces a matrix B in reduced row{

echelon form, which is row{equivalent to A. If A is the augmented matrix

of a system of linear equations, then B will be a much simpler matrix than

A from which the consistency or inconsistency of the corresponding system

is immediately apparent and in fact the complete solution of the system can

be read o®.

STEP 1.

Find the ¯rst non{zero column moving from left to right, (column c1)

and select a non{zero entry from this column. By interchanging rows, if

necessary, ensure that the ¯rst entry in this column is non{zero. Multiply

row 1 by the multiplicative inverse of a1c1 thereby converting a1c1 to 1. For

each non{zero element aic1 ; i > 1, (if any) in column c1, add ¡aic1 times

row 1 to row i, thereby ensuring that all elements in column c1, apart from

the ¯rst, are zero.

STEP 2. If the matrix obtained at Step 1 has its 2nd; : : : ;mth rows all

zero, the matrix is in reduced row{echelon form. Otherwise suppose that

the ¯rst column which has a non{zero element in the rows below the ¯rst is

column c2. Then c1 < c2. By interchanging rows below the ¯rst, if necessary,

ensure that a2c2 is non{zero. Then convert a2c2 to 1 and by adding suitable

multiples of row 2 to the remaing rows, where necessary, ensure that all

remaining elements in column c2 are zero.

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