There exist real invertible matrices A and S such that .
Therefore, the given statement is not satisfied.
On taking determinant
Which is false, as det(S) turns out to be imaginary.
The question is whether there exists an invertible matrix A such that A and 2A are similar. If they were similar, it would have to apply
det A = det(-A)
det A = - detA,
Which, since , is a contradiction. Therefore, the given statement is not satisfied.
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