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For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by$R(M)$ the minimal rank of all the matrices obtained by changing some numberson the main diagonal of $M$. We prove that for each non-negative integer $k$there is a polynomial in $n$ algorithm deciding whether $R(M) \leq k$ (whosecomplexity may depend on $k$). We also give a polynomial in $n$ algorithmcomputing a number $m$ such that $m/2 \leq R(M) \leq m$. These results haveapplications to graph drawings on non-orientable surfaces.

On the rank of Z_2-matrices with free entries on the diagonal