Eigenvalues for a matrix A satisfying A^2=-A
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Let A be an nxn matrix such that A^2=-A. Then the possible values of the eigenvalues of A are
i) 0, ii) 1, iii)-1, iv) i, v) -i
A. i) and ii) only
B. i) and iii) only
C. iv) and v) only
D. i), iii), iv), and v)
E. All of these
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Solution Summary
We study the possible eigenvalues of an nxn matrix A that satisfies A^2=-A.
Solution Preview
If a is a (possibly complex) eigenvalue of A, and v is an eigenvector associated to the eigenvalue a, then
Av=av.
Multiply by A:
A^2 v=a Av=a(av)=a^2 v
And A^2=-A ...
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