how can we prove that the subset of all invertible matrices is open?

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Why do the $n \times n$ non-singular matrices form an “open” set?

Consider the space of all nxn matrices with real entries with the standard metric, i.e.,view the matrix as an element of $R^{n^2} $and use the usual Euclidean metric on $R^{n^2} $. I need to prove that the subset of all invertible matrices is open. Please any idea?

Another related question: http://math.stackexchange.com/questions/18964/what-about-gln-mathbb-c-is-it-open-dense-in-mn-mathbb-c — Jonas Meyer, Feb 18 '12 at 08:54

score 5 · Accepted Answer · answered Feb 18 '12 at 06:35

5

$\bf Hint:$ $A$ is invertible iff the determinant is different from zero.

answered Feb 18 '12 at 06:35

azarel

13,120

how can we prove that the subset of all invertible matrices is open?

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