There is an alternative criterion for positive definiteness based on subdeterminants of the matrix. The 2x2 version already appears in (3.62).
(a) Prove that a 3 x 3 matrix

Is positive definite if and only if α > 0, ad - b2 > 0 det K > 0.
(b) Prove the general version: an n x n matrix K > 0 is positive definite if and only if all the upper left square k x k subdeterminants are positive for k = 1 ,........., n Hint: See Exercise 1.9.19

abc