Prove that $0<\det(A) \le 1$

$A=(a_{ij})$ is a $n\times n$ symmetric real matrix such that:

$a_{ii}=1$ and $\sum_{j=1}^{n}|a_{ij}|<2$ for all $i \in \{1,2,3,...,n\}$.

Prove that $0< \det(A) \le 1$.

My approach:

That is a question that I have tried before and I am trying again but still without success.

I'm trying to use spectral theorem (maybe prove that $|\lambda| \le1$) but I got nothing.

I also tried brute force using the definition (using permutations) of $\det A$.

Any idea?

All eigenvalues are real, this combined with the gershgorin disk theorem implies every eigenvalue is in the range $(0,2)$. So the determinant is positive.

Notice that the trace of the matrix is $n$, and this is equal to the sum of the eigenvalues, so by the arithmetic-geometric mean the product of the eigenvalues is at most $1$.