Min/Max/Saddle Points In Multivariable Function

Question

I am learning about Min/Max/Saddle points and I am trying to make some order in what I have learned.

So when we have a multivarible function to find Min/Max/Saddle points:

1. we take the gradient $\nabla f=0$

2. we take the second order derivatives and create the hessian matrix

3. we plug in the values we found in 1

If the hessian matrix is 2 by 2 of the form $H=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

then if $det(H)>0$: 1. if $a>0$: minimum 2. if $a<0$: maximum

if $det(H)<0$ saddle point if $det(H)=0$ we can determine

If the hessian matrix is 3 by 3 we take:

a. the upper left 1 by 1 minor

b. the upper left 2 by 2 minor

c. the upper left 3 by 3 minor

If all are positive it is a minimum

If the change sign + - + or - + - it is maximum

When will it be a saddle point in the 3 by 3 case? is there a reference to a fully detailed summery of the process?