Math 2331
Linear Algebra
Section 2.6
Elimination and Factoization A=LU
Elimination -
Goal: Reduce a matrix A to an upper triangular matrix U so that Ax=b may be solved by
back substitution with U. The process of elimination may be represented as multiplication
by invertible elimination matrices (from the left). Let's first look at these matrices
Legal Elimination Operations
Let's work with the matrix
1
3
2
2
2
2
3
5
7
A
⎛
⎞
⎜
⎟
=
⎜
⎟
⎜
⎟
⎝
⎠
1. Replace Row i by Row i minus a multiple c of row j.
This matrix is
and this is the identity matrix plus the number -c in the i,j position
ij
E
To replace row 2 with row 2 - 2 row 1 use the matrix
21
1
0
0
2
1
0
0
0
1
E
⎛
⎞
⎜
=
−
⎜
⎜
⎟
⎝
⎠
⎟
⎟
⎟
⎟
times A =
To replace row 3 with row 3 - 3 row 1 use the matrix
31
1
0
0
0
1
0
3
0
1
E
⎛
⎞
⎜
=
⎜
⎜
⎟
−
⎝
⎠
times A =
31
21
E E
=
Times A =

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