Can a symmetric, positive-definite, real matrix with only 1s on the main diagonal have an off diagonal element with absolute value greater than 1?

Question

I am working with correlation matrices and I would like to know if every symmetric, positive definite matrix with 1s on the main diagonal is a correlation matrix (i.e. all its off diagonal elements have absolute value less than 1). This is obviously true in dimension 2 and, till now, I couldn't find a counterexample in higher dimensions so I suspect that the answer to the main question is NO but I couldn't prove it.

Thanks in advance.

Answer 1

HINT: For any $n\times n$ real SPD matrix $A$ and $1\leq i,j\leq n$, $a_{ij}^2\leq a_{ii}a_{jj}$ (because all principal submatrices of an SPD matrix are SPD).