# The Number of symmetric,PD, $8\times 8$ matrices

The Number of symmetric,Positive Definite, $8\times 8$ matrices having trace$=8$ and determinant$=1$ is

1. $0$

2. $1$.

3. $>1$ but finite.

4. $\infty$

I am not able to do this one.

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 Hint: diagonal matrices. – Gerry Myerson Jul 8 '12 at 23:03 Yes, so Only $1$ – Taxi Driver Jul 8 '12 at 23:17

If $A$ is pos. def. then its eigenvalues $\lambda_i$ are real and positive. Besides, we know (don't we?) that, for any matrix, $\sum \lambda_i = tr(A)$ and $\prod \lambda_i= |A|$. In our case, that means that we are restricted to $\sum \lambda_i =8$ and $\prod \lambda_i =1$... (can you go on from here? hint: arithmetic-geometric means and their properties)