# 3x2 matrices in RREF

1. Apr 21, 2015

### pyroknife

1. The problem statement, all variables and given/known data
How many 3x2 matrices are there in RREF form?

2. Relevant equations

3. The attempt at a solution
I counted 5, but the solutoin in my book says 4.

0 0
0 0
0 0

1 0
0 1
0 0

1 1
0 0
0 0

0 1
0 0
0 0

1 0
0 0
0 0

2. Apr 21, 2015

### Staff: Mentor

Think about these matrices as a system of three equations in two unknowns, x and y.
ax + by = 0
cx + dy = 0
ex + fy = 0
This system of equations represents three lines in the plane. The 2nd reduced matrix you show above says that all three lines intersect at a single point, the origin - (0, 0). What do the other matrices represent geometrically?

3. Apr 21, 2015

### pyroknife

The 0 zero matrix represents of all of the plane. The 3rd, 4th, and 5th matrix represent some line on the plane.

Aren't all 5 of those matrices in RREF form though?

4. Apr 21, 2015

### pyroknife

Oh wait I see.

1 0
0 0
0 0

1 1
0 0
0 0

are essentially the same since that second element (top right) can be anything (e.g., 0, 1, .....)

5. Apr 21, 2015

### Staff: Mentor

No, I don't think so. The first matrix says x = 0, so all three lines lie along the y axis. The second matrix says x + y = 0, or y = -x, so all three lines lie along this line.

6. Apr 21, 2015

### Staff: Mentor

What would the system have to look like so that you ended up with the first matrix (the one with all zeroes)?

7. Apr 21, 2015

### pyroknife

Wouldn't the system just be zero?

I can't think of any matrices that would row reduce to 0 besides 0 itself.

8. Apr 21, 2015

### Staff: Mentor

The system wouldn't be 0, it would have to look like this, wouldn't it?
0x + 0y = 0
0x + 0y = 0
0x + 0y = 0
My thought is that they aren't considering this as a legitimate system of equations.

9. Apr 21, 2015

### pyroknife

Oh, hmmm. I think my book considers the 0 matrix as in RREF.

And I'm a little confused with the semantics here. Why is the system not zero? All I see are zeros.

I think my post #4 is correct, since we can put any # in that top right element:
i.e.,
1 c
0 0
0 0
where c can be anything.

10. Apr 21, 2015

### Staff: Mentor

One other thought - what you show as
1 1
0 0
0 0

should probably look more like
a b
0 0
0 0
Edit:
1 c
0 0
0 0
would work as well.

Your (1 1) version provides for the possibility that the three lines all lie on the graph of y = -x. My version allows for all possible non-vertical, non-horizontal lines.

For me, it's helpful to look at the geometry -- the system of three equations in two variables represents three lines that all go through (0, 0), since each equation is of the form ax + by = 0. Since they all go through (0, 0) it's not possible that we have a system where one or two lines are parallel to the third, hence no solution. So we have to have one of the following:

All three lines are vertical -- x = 0, your 4th matrix
All three lines are vertical -- y = 0, your 5th matrix
All three lines intersect at the origin -- your 2nd matrix
All three lines lie on the same non-vertical, non-horizontal line -- my matrix above

As far as your matrix of all zeroes, that's a legitimate matrix, but I don't see how it could be the result of any non-trivial system of equations.

11. Apr 21, 2015

### Staff: Mentor

Be careful to distinguish between the number 0 and the $\mathcal{0}$ matrix. Here it should be understood that we're talking about matrices, all of whose elements are zero.

Response to your question below: A system consists of one or more equations. Equations are different from numbers or matrices. There is no notation that I know of to represent a "zero" system. What I'm saying is, don't describe a system as "zero", but I guess you could talk about a system of equations where all of the coefficients are zero.