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Multiply row 3 by –3, add it to row 1, replacing it:
3\#3#1
@RCIJ!
Multiply row 2 by –2, add it to row 1, replacing it:
2\#2#1
@RCIJ!
Writing this process by hand will result in the following steps:
≅
≅
−
−
=
2
1
7
1
0
0
1
1
0
3
2
1
2
3
7
1
0
0
1
1
0
3
2
1
14
3
7
7
0
0
1
1
0
3
2
1
aug
A
.
2
1
1
1
0
0
0
1
0
0
0
1
2
1
1
1
0
0
0
1
0
0
2
1
−
≅
≅
aug
A
Pivoting
If you look carefully at the row operations in the examples shown above, you
will notice that many of those operations divide a row by its corresponding
element in the main diagonal. This element is called a pivot element, or
simply, a
pivot
. In many situations it is possible that the pivot element
become zero, in which case we cannot divide the row by its pivot. Also, to
improve the numerical solution of a system of equations using Gaussian or
Gauss-Jordan elimination, it is recommended that the pivot be the element
with the largest absolute value in a given column. In such cases, we
exchange rows before performing row operations. This exchange of rows is
called
partial pivoting
. To follow this recommendation is it often necessary to
exchange rows in the augmented matrix while performing a Gaussian or
Gauss-Jordan elimination.
While performing pivoting in a matrix elimination procedure, you can
improve the numerical solution even more by selecting as the pivot the element
with the largest absolute value in the column and row of interest. This
operation may require exchanging not only rows, but also columns, in some
