how to show that a nonsingular positive semidefinite matrix is positive definite?

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Got a question regarding the following statement: a nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how to show it.

If the above statement is not true, how about this statement: a symmetric nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how to show it.

Thanks!