# Product of Diagonal Matrices

1. Apr 28, 2012

### jsgoodfella

If we take an nxn diagonal matrix, and multiply it by an nxn matrix C such that AC=CA, will C be diagonal? I know, for instance, if C is a matrix with ones in every entry, AC=CA holds. But is there a more general way to format such a counterexample, or have I already provided a sufficient "proof"?

Thanks in advance. This isn't a homework question.

2. Apr 28, 2012

### DonAntonio

$$\left(\begin{array}{cc}2&0\\0&2\end{array}\right) \left(\begin{array}{cc}1&1\\0&1\end{array}\right)=\left(\begin{array}{cc}2&2\\0&2\end{array}\right)=\left(\begin{array}{cc}1&1\\0&1\end{array}\right) \left(\begin{array}{cc}2&0\\0&2\end{array}\right)$$
DonAntonio

3. Apr 28, 2012

### phyzguy

When you disprove a statement by providing a counter-example, one counter-example is sufficient. There is no need to provide more.

4. Apr 28, 2012

### DonAntonio

Yes, of course. Whom are you addressing and why?

DonAntonio

5. Apr 28, 2012

### phyzguy

The OP asked whether he had already provided a sufficient proof. The answer is yes - since had already provided one counter-example, this is sufficient to disprove the original statement. That's all I'm saying.

6. Apr 28, 2012

### DonAntonio

Good. This time you provide a quote of whom you're addressing and thus we know. The last time we, or at least I, didn't know.

DonAntonio