Eigenvalues of a 2 × 2 matrix A such that A 2 = II have...

oxidricasbt7

oxidricasbt7

Answered

2022-12-20

Eigenvalues of a 2 × 2 matrix A such that A 2 = I
I have no idea where to begin.
I know there are a few matrices that support this claim, will they all have the same eigenvalues?

Answer & Explanation

Kendall Cortez

Kendall Cortez

Expert

2022-12-21Added 4 answers

If v is an eigenvector of A with eigenvalue λ, then
v = I v = A 2 v = A ( A v ) = A ( λ v ) = λ ( A v ) = λ 2 v .
Thus, if λ is an eigenvalue of A and A 2 = I then λ 2 = 1. This gives only two possibilities for λ, ± 1.
Notice, that we never assumed that A is 2 × 2. Indeed, if A is any square matrix and A 2 = I then the only possible eigenvalues of A are ± 1.

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