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If you were performing these operations by hand, you would write the
following:
−
−
−
−
≅
−
−
−
−
=
4
3
7
1
2
4
1
2
3
3
2
1
4
3
14
1
2
4
1
2
3
6
4
2
aug
A
−
−
−
≅
−
−
−
−
−
−
≅
32
3
7
13
6
0
1
1
0
3
2
1
32
24
7
13
6
0
8
8
0
3
2
1
aug
A
−
−
≅
14
3
7
7
0
0
1
1
0
3
2
1
aug
A
The symbol
≅
(“ is equivalent to”) indicates that what follows is equivalent to
the previous matrix with some row (or column) operations involved.
The resulting matrix is upper-triangular, and equivalent to the set of equations
X +2Y+3Z = 7,
Y+ Z = 3,
-7Z = -14,
which can now be solved, one equation at a time, by backward substitution,
as in the previous example.
Gauss-Jordan elimination using matrices
Gauss-Jordan elimination consists in continuing the row operations in the
upper-triangular matrix resulting from the forward elimination process until an
identity matrix results in place of the original
A
matrix. For example, for the
case we just presented, we can continue the row operations as follows:
Multiply row 3 by –1/7:
7\Y 3
@RCI!
Multiply row 3 by –1, add it to row 2, replacing it:
1\ #
3
#2
@RCIJ!
