Give an example of an \(n \times n\) matrix which has all its principal minors positive, but which is not a positive matrix.
Solution
Take \[ A= \begin{pmatrix} 1&1\\ 0&1 \end{pmatrix}. \] Its principal \(1\times1\) minors are both equal to \(1\), and its determinant is \[ \det A=1. \] Thus every principal minor is positive.
However, \[ A^*= \begin{pmatrix} 1&0\\ 1&1 \end{pmatrix} \ne A. \] So \(A\) is not Hermitian. Since every positive matrix must be Hermitian, \(A\) is not positive.
Hence positivity of all principal minors alone does not imply positivity of a matrix.