Theorem 7.10 (6.10) says ``We leave a proof of this result to
the student.'' Well, you are the student.
Hint: there are at least two different ways to do this:
- You can note that inverses of elementary row operations are
elementary row operations. You can use this to show that if
can be reduced to two different row canonical matrices, then one
of the two can be reduced to the other using elemenary row
operations. Then you can study what can change or not in doing
this.
- You can look at the most general solution of the homogeneous
system of equations given by matrix . This solution will not
change by elementary row operations. So if there are two row
canoncical matrices, they must still have the same homogeneous
solution. Then you can study what is needed to get the same
homogeneous solution.
Explain your reasoning clearly.