# Linear algebra: Matrices Question

1. Sep 24, 2009

### nietzsche

1. The problem statement, all variables and given/known data

Show that there are no 2 x 2 matrices A and B such that

AB - BA =
$$\left( \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right)$$

2. Relevant equations

The matrix is the 2 x 2 identity matrix.

3. The attempt at a solution

I tried to use variables such as a(1 1) a(1 2)... and then I did the multiplication and subtraction. When I set each entry equal to the entries in the identity matrix, I ended up with so many variables that I didn't know what do with all of them. None of them seemed to match up or eliminate. Is there an easier way to do this?

2. Sep 24, 2009

### Dick

Yes, there is an easier way. Take the trace of both sides. Look up some of the properties of the trace.

3. Sep 24, 2009

### nietzsche

My instructor didn't teach us anything about the trace, so I don't think we're supposed to use it. Are there any other ways? Is it even possible the way that I was doing it?

My instructor could have just assigned the question without realizing that it required other knowledge.

4. Sep 24, 2009

### Dick

If you want to do it in a really basic way, write out the product AB-BA in terms of the individual entries. You may have already done this. Can you show the sum of the two diagonal entries is 0? (That's exactly what the trace would tell you.)

5. Sep 24, 2009

### nietzsche

Ah, thank you very much, I see now.