In , problem 5 the author shows that the differential of the determinant map $d(\det)_A$ for an invertible matrix $A$ is nonsingular by only showing that $d(\det)_A(A) \ne 0$. I don't really see why this shows that $d(\det)_A$ is nonsingular. I would like some further clarification on this point.
Need help understanding proof about critical values of the determinant map.
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multivariable-calculus
differential-topology