A First Course in Linear Algebra, 2017a

2.1. Matrix Arithmetic 85
→
⎡
⎣
1 0 0 1 0
0 1 1 0 0
0 0 0 −2 1
⎤
⎦
The left side of this matrix is B, and the right side is U. Comparing this to the matrix U found above
in Example 2.54, you can see that the same matrix is obtained regardless of which process is used. ♠
Recall from Algorithm 2.37 that an n × n matrix A is invertible if and only if A can be carried to the
n×n identity matrix using the usual row operations. This leads to an important consequence related to the
above discussion.
Suppose A is an n × n invertible matrix. Then, set up the matrix [A|I n ] as done above, and row reduce
until it is of the form [B|U]. In this case, B = I n because A is invertible.
B = UA
I n = UA
U −1 = A
Now suppose that U = E 1 E 2 ···E k where each E i is an elementary matrix representing a row operation
used to carry A to I. Then,
U −1 =(E 1 E 2 ···E k ) −1 = Ek
−1 ···E2 −1 E−1 1
Remember that if E i is an elementary matrix, so too is Ei
−1 . It follows that
A = U −1
= Ek
−1 ···E2 −1 E−1 1
and A can be written as a product of elementary matrices.
Theorem 2.57: Product of Elementary Matrices
Let A be an n × n matrix. Then A is invertible if and only if it can be written as a product of
elementary matrices.
Consider the following example.
Example 2.58: Product of Elementary Matrices
⎡
0 1
⎤
0
Let A = ⎣ 1 1 0 ⎦. Write A as a product of elementary matrices.
0 −2 1
Solution. We will use the process outlined in Theorem 2.55 to write A as a product of elementary matrices.
We will set up the matrix [A|I] and row reduce, recording each row operation as an elementary matrix.