Suppose $A$ is symmetric, positive semidefinite and all its diagonal entries are strictly positive (real coefficients - even integer if it helps). Suppose that the first $r$ rows of $A$ are linearly independent.

Is it true that the $r$-th leading principal submatrix is full rank (and therefore positive definite)?

(The $r$-th leading principal submatrix is the top-left $r\times r$ submatrix of $A$ - it is certainly positive semidefinite and symmetric)

What if all the entries are strictly positive?

I asked this on math.stackexchange last week but I did not receive any suggestion: https://math.stackexchange.com/questions/2221146/full-rank-submatrices-of-positive-semidefinite-matrix-with-positive-entries