tell Positive Definite Matrices by the sign of determinants

Question

This is from my textbook

I don't understand why it didn't mention the other situation which is $det(A_k) <0 $ for all k, and we stall have positive pivot because $\frac{negative}{negative}=positive$

Answer 1

If you take that equation seriously for $k=1$, you need to evaluate $\det A_0$, the determinant of an empty matrix. That's the sum over an empty product for all permutations of $0$ elements, of which there is $1$, so $\det A_0=1$. Thus the case $\det A_k\lt0$ is excluded. A more direct way might be to consider that equation to hold for $k\gt1$ and regard the first pivot $a_{11}=\det A_1$ as an initial value.