# Linear algebra: Matrices Question

## Homework Statement

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

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

## Homework Equations

The matrix is the 2 x 2 identity matrix.

## 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?

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

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

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.

Dick
Homework Helper

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.

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.)

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.)

Ah, thank you very much, I see now.