Invertibility / singularity of matrix resulting from a symmetric(0,1) matrix pointwise multiplicated with a vector (consisting of a,b)+identity matrix

I am trying to find conditions then a certain matrix is invertible or not (which is problematic as the matrix is random). The matrix results from the following: $A=\Tilde{A}+diag(n)$. Furthermore $\Tilde{A}$ results from the pointwise multiplication of a random symmetric matrix (consisting of 0 and 1, but necessarily 0 on the diagonal) with a random vector which constsis of $\alpha$ and $\beta$ entries.

Does anyone have any ideas how to deduce some criterions for the invertibility of matrix $A$?

Thank you so much!

I already tried thinking about LU decomoposition, but could not deduce any criterion. Obviosly, it fully depends on how the random matrices look and linear dependence between the rows are less likely if the dimension is higher...

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