154 CHAPTER 8. RANK OF A MATRIX

The following sequence of row operations leads to the row reduced echelon form 1 1 3 11 2 5 31 −1 −1 −3

→ 1 1 3 1

0 1 2 21 −1 −1 −3

→ 1 1 3 1

0 1 2 20 −2 −4 −4



→

 1 1 3 10 1 2 20 0 0 0

→ 1 0 1 −1

0 1 2 20 0 0 0

Thus the sequence of elementary matrices is 1 −1 0

0 1 00 0 1

 1 0 0

0 1 00 2 1

 1 0 0

0 1 0−1 0 1

 1 0 0−1 1 00 0 1

From the way we are multiplying on the left the matrices must be listed in this order.

Example 8.2.14 Here is an invertible matrix. A =

 1 2 11 3 00 −1 2

 . Express A as a prod-

uct of elementary matrices.

Here are a sequence of row operations which row reduce A to the identity. 1 2 11 3 00 −1 2

→ 1 2 1

0 1 −10 −1 2

→ 1 2 1

0 1 −10 0 1

→ 1 2 1

0 1 00 0 1

→ 1 2 0

0 1 00 0 1

→ 1 0 0

0 1 00 0 1

Thus the sequence of elementary matrices which produces the row reduced echelon form,in this case, the identity matrix is the following product multiplied on the left times A : 1 −2 0

0 1 00 0 1

 1 0 −1

0 1 00 0 1

 1 0 0

0 1 10 0 1

 1 0 0

0 1 00 1 1

 1 0 0−1 1 00 0 1

Thus, to obtain A, we replace each of these with its inverse and multiply the result in theopposite order to get A as indicated in Proposition 8.2.12. Thus we want 1 0 0

1 1 00 0 1

 1 0 0

0 1 00 −1 1

 1 0 0

0 1 −10 0 1

 1 0 1

0 1 00 0 1

 1 2 0

0 1 00 0 1

You can check that the product of these does equal A.