Elementary Row Operations, REF, and RREF
Now that we know a little bit about matrices, we’re going to learn how to use matrices to solve
problems!
One of the most useful things we can do to a matrix is to “row reduce” it.
Row reduction is a process by which a matrix is simplified into an “equivalent” matrix which is
easier to use overall. In order to make this procedure more canonical, we’ll perform our reduction
using a very precise collection of operations known as elementary row operations.
Throughout, we’ll refer to the matrix
M=
4 1 −2 0
1 0 0 −5
0 1 3 −1
1 1 −1−1
for all of our examples.
Elementary Row Operations
There are three elementary row operations that we can perform on a matrix to get a new matrix
which is considered “row equivalent” to it:1
1. Interchange two rows.
Example: Interchanging rows 2 and 4 (shorthand: R2 ↔R4) in Myields the following:
4 1 −2 0
1 0 0 −5
0 1 3 −1
1 1 −1−1
R2↔R4
−−−−→
4 1 −2 0
1 1 −1−1
0 1 3 −1
1 0 0 −5
2. Multiply all entries in a row by a nonzero constant.
Example: We can multiply row 1 of Mby 3 (shorthand: R1 = 3R1) to get the following:
4 1 −2 0
1 0 0 −5
0 1 3 −1
1 1 −1−1
R1=3R1
−−−−→
12 3 −6 0
1 0 0 −5
0 1 3 −1
1 1 −1−1
1Definition: Two matrices Mand Nare said to be row equivalent if there is a series of elementary row operations
which transforms Minto N(and vice versa).
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