The statement isn't true, you have to make additional assumptions. E.g. one could add the condition that all the matrix elements are real. Then |A|^3 =1 and you know that |A| must be a real number, so |A| must be 1.
The condition that the matrix elements are real is sufficient, but not necessary. E.g., take the 2x2 diagonal matrix with exp(2 pi i/3) and
exp(-2 pi i/3) on the diagonal.
I'm having trouble with doing this matrix proof
The question is "Given some matrix A has the property A*2=A^-1, show that determinant A = 1, i.e |A| = 1"
I've tried for ages, but I can't seem to do it, this is what I got to
A^2= A^-1
|A^2| = |A^-1|
|A|^2 = 1/|A|
Can someone please help?
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