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Q32 E

Expert-verifiedFound in: Page 309

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**There exist real invertible** ${\mathbf{3}}{\mathbf{\times}}{\mathbf{3}}$ **matrices** **A****and** *S***such that** **. **

Therefore, the given statement is not satisfied.

On taking determinant

$det\left(S\right)det\left(A\right)det\left({S}^{T}\right)={\left(-1\right)}^{3}det\left(A\right)\phantom{\rule{0ex}{0ex}}-{\left(det\left(S\right)\right)}^{2}detA=det\left(A\right)$,

Which is false, as det(S) turns out to be imaginary.

The question is whether there exists an invertible $3\times 3$ 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 $detA\ne 0$, is a contradiction. Therefore, the given statement is not satisfied.

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