# Positive-definitiveness of non-negative matrix with only positive diagonal values?

I'm trying to assess and prove whether a matrix containing only non-negative values (generated from the standard normal distribution) and diagonal values which are always positive (obtained as a product of the identity matrix and constants: $\alpha$ (size of the matrix)) and $\beta$ (a real number) is positive definite.

The matrix might be composed of two other matrices that easily create these results. For example: A multiplied by its transpose summed with the identity matrix.

$AA' + \alpha\beta I$

This example would show the matrix to be symmetric, so we also need to show the positive definiteness

Is there some proof relating to the properties that allows a positive definite proof to be created without proving all eigenvalues to be positive for example?

EDIT: Originally forgot to specify that values of the matrix are from the standard normal distribution and there is a multiplier on the diagonal using the product of the length of the matrix and another real parameter

No, this won't necessarily happen, for fairly basic reasons: for example, $$\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$$ has negative determinant, so it can't be positive-definite.