Page 42 - 35Linear Algebra

42 Systems of Linear Equations

Example 15 (Silly order of equations)
A robot might make this mistake:

) !
0x + y = −2 0 1 −2
⇔ ∼ · · · ,
x + y = 7 1 1 7
and then give up because the the upper left slot can not function as a pivot since the 0
that lives there can not be used to eliminate the zero below it. Of course, the right
thing to do is to change the order of the equations before starting
) ! ! (
x + y = 7 1 1 7 1 0 9 x + 0 = 9
⇔ ∼ ⇔
0x + y = −2 0 1 −2 0 1 −2 0 + y = −2 .

The third augmented matrix above is the RREF of the first and second. That is to
say, you can swap rows on your way to RREF.

For larger systems of equations redundancy and inconsistency are the ob-
structions to obtaining the identity matrix, and hence to a simple statement
of a solution in the form x = a, y = b, . . . . What can we do to maximally
simplify a system of equations in general? We need to perform operations
that simplify our system without changing its solutions. Because, exchanging
the order of equations, multiplying one equation by a non-zero constant or
adding equations does not change the system’s solutions, we are lead to three
operations:

• (Row Swap) Exchange any two rows.

• (Scalar Multiplication) Multiply any row by a non-zero constant.

• (Row Addition) Add one row to another row.

These are called Elementary Row Operations, or EROs for short, and are
studied in detail in section 2.3. Suppose now we have a general augmented
matrix for which the first entry in the first row does not vanish. Then, using
1
just the three EROs, we could then perform the following.

1
This is a “brute force” algorithm; there will often be more efficient ways to get to
RREF.

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