Filter
Top Products
Page 11-33
Multiply row 2 by –1/8: 8\Y2 @RCI!
Multiply row 2 by 6 add it to row 3, replacing it: 6#2#3 @RCIJ!
If you were performing these operations by hand, you would write the
following:
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:
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−
−
−≅
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−
−
−=
4
3
7
124
123
321
4
3
14
124
123
642
aug
A
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−−−
≅
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
−
−−
−−≅
32
3
7
1360
110
321
32
24
7
1360
880
321
aug
A
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−−
≅
14
3
7
700
110
321
aug
A