Trace of 0x0 matrix.

Question

A lot of questions on the site ask what the determinant of a $0\times 0$ matrix should be. It seems to be well agreed that it should be $1$. What about the trace? I think it should be $0$ since the empty sum is $0$.

Answer 1

Yes, given square matrices, $A,B$ with $$C=\begin{pmatrix}A&0\\0&B\end{pmatrix}$$

You want $\det C=\det A\det B$ and $\operatorname{tr} C= \operatorname{tr}A+ \operatorname{tr} B.$

If $A$ is $0\times 0,$ you get $C=B.$

And yes, this is the same reason you define empty products as $1$ and empty sums as $0.$