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.