Question

a) Prove that the set of invertible real 2 × 2 matrices is dense in the set of all real 2 × 2 matrices.

b) The set of diagonalisable 2 × 2 matrices dense in the set of all real 2 × 2 matrices.

we can use a proof or an counterexample...

Answer 1

I don't know how to say when it 's trivial like that. Let think

Answer 2

they can overlap, too.

Answer 3

@dan815 help me dan, I don't know how to explain

Answer 4

mmm. how do we prove that without pictures? good visual aid though. i like it.

Answer 5

Recall that for a general \(2\times2\) matrix \(\mathbf{A}\):$$\mathbf{A}=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$for \(a,b,c,d\in\mathbb{R}\) to be invertible it must have a non-zero determinant (i.e. no eigenvalues equal to \(0\) and therefore an empty nullspace :-). Our determinant is just \(\det\mathbf{A}=ad-bc\) so we're interested in \(ad-bc=0\Leftrightarrow ad=bc\). It makes sense that there are comparatively few combinations of \(a,b,c,d\) out of \(\mathbb{R}\) that satisfy the above equation. Therefore we can say with certainty invertible matrices comprise almost all \(2\times2\) real matrices and are therefore dense in the set

Answer 6

what metric are you using?

Answer 7

what do you mean metric?