220 CHAPTER 10. A FEW FACTORIZATIONS
Review how to take the inverse of an elementary matrix. Then we can do the follow-ing.
A =
1 0 02 1 00 0 1
1 0 0−2 1 00 0 1
1 2 3
2 0 −21 3 1
=
1 0 02 1 00 0 1
1 2 3
0 −4 −81 3 1
Next
A =
1 0 02 1 00 0 1
1 0 0
0 1 01 0 1
1 0 0
0 1 0−1 0 1
1 2 3
0 −4 −81 3 1
=
1 0 02 1 01 0 1
1 2 3
0 −4 −80 1 −2
Next
A =
1 0 02 1 01 0 1
1 0 0
0 1 00 −1/4 1
1 0 0
0 1 00 1/4 1
1 2 3
0 −4 −80 1 −2
=
1 0 02 1 01 − 1
4 1
1 2 3
0 −4 −80 0 −4
Using this example, describe why, when a matrix can be reduced to echelon formusing only row operation 3, then it has an LU factorization.