Is a symmetric, invertible and positive semi-definite matrix automatically positive definite?

Question

My question is: Is a symmetric, invertible and positive semi-definite matrix automatically positive definite?

Any solution or hints on how this can be answered would be appreciated.

Answer 1

Note that for a positive semi-definite matrix ($\lambda_i\ge 0$) and invertible ($\lambda_i\neq 0$) we have that $\lambda_i> 0$ then the matrix is positive definite.