invertible to positive absolute link

why if a matrices is not invertible then its not absolute positive
?

positive-definite matrices cannot be singular. suppose so, then for a positive-definite matrix A, there is some x ≠ 0, with Ax = 0. hence x^T(Ax) = 0, contradicting the fact that A is positive-definite.

since a positive-definite matrix is non-singular, it posseses an inverse, so any non-invertible matrix cannot possibly be positive-definite.

yes i understand now
if its definite positive then all of its eigenvalues are positive
T(v)=kv the only way for T(v)=0 is that v=0
so the kerT={0}
so its invertible
but our A is not invertible so its not definite positive
correct?

thanks:)

that is a valid argument also.