Mathematics Asked by user469754 on November 27, 2020

My attempt: I take the matrices which have trace $-1$ and determinant $1$.

As I have tried this question many times and I did not get any matrices which satisfy the conditions of option a), option b), or option c), so from my point of view none of the options are correct.

Any help is appreciated.

Here are some matrices to think about.

$A = pmatrix {1\&1\&&-1}$

$A = pmatrix {cos frac {2pi}{3}& sin frac {2pi}{3}\-sin frac {2pi}{3}&cos frac {2pi}{3}}$

$A = pmatrix {1&&\&cos frac {2pi}{3}& sin frac {2pi}{3}\&-sin frac {2pi}{3}&cos frac {2pi}{3}}$

In all of the scenarios

$A^n = I$ says something about the eigenvalues of $A$

Then there is a further clause that restricts what those eigenvalues may be.

And finally, do all matrices that meet the previous 2 constraints meet the 3rd constraint.

Update.

$A = pmatrix {-frac 12& frac {sqrt 3}{2}\-frac {sqrt 3}{2}&-frac 12}$

$A^2 = pmatrix {-frac 12& -frac {sqrt 3}{2}\frac {sqrt 3}{2}&-frac 12}$

$A^2+A+I = 0$

Answered by Doug M on November 27, 2020

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