A First Course in Linear Algebra, 2017a

2.1. Matrix Arithmetic 83
This equals I so we know that we have computed E −1 properly.
♠
Suppose an m × n matrix A is row reduced to its reduced row-echelon form. By tracking each row
operation completed, this row reduction can be completed through multiplication by elementary matrices.
Consider the following definition.
Definition 2.53: The Form B = UA
Let A be an m × n matrix and let B be the reduced row-echelon form of A. Then we can write
B = UA where U is the product of all elementary matrices representing the row operations done to
A to obtain B.
Consider the following example.
Example 2.54: The Form B = UA
⎡ ⎤
0 1
Let A = ⎣ 1 0 ⎦. FindB, the reduced row-echelon form of A and write it in the form B = UA.
2 0
Solution. To find B, row reduce A. For each step, we will record the appropriate elementary matrix. First,
switch rows 1 and 2.
⎡ ⎤ ⎡ ⎤
0 1 1 0
⎣ 1 0 ⎦ → ⎣ 0 1 ⎦
2 0 2 0
The resulting matrix is equivalent to finding the product of P 12 = ⎣
Next, add (−2) times row 1 to row 3.
⎡
⎣
1 0
0 1
2 0
⎤
⎡
⎦ → ⎣
1 0
0 1
0 0
⎤
⎦
⎡
0 1 0
1 0 0
0 0 1
This is equivalent to multiplying by the matrix E(−2 × 1 + 3) = ⎣
resulting matrix is B, the required reduced row-echelon form of A.
We can then write
B = E(−2 × 1 + 2) ( P 12 A )
It remains to find the matrix U.
= ( E(−2 × 1 + 2)P 12) A
= UA
U = E(−2 × 1 + 2)P 12
⎡
⎤
⎦ and A.
1 0 0
0 1 0
−2 0 1
⎤
⎦. Notice that the