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a) Identify all possible eigenvalues of an $$n \times n$$ matrix $$A$$ that satisfies the matrix equation: $$A-2 I=-A^{2}$$. Justify your answer.
b) Must $$A$$ be invertible?
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a) Multiplying the equation $$A-2 I=-A^2$$ by $$A+2 I$$ on both sides, we have
$A^2 + 4A - 4I = -A^3 - 2A^2 = (A + 2I)(-A^2) = -A^3 - 4A^2 - 4A = 0.$
Since $$A^2+4 A-4 I=0, A^3+4 A^2+4 A-4 I=0$$.
Hence, the characteristic polynomial of $$A$$ is $$p(x)=x^3+4 x^2+4 x-4$$.
The roots of the polynomial can be found by factoring, or using the cubic formula.
In either case, the three roots are $$\mathrm{x}=1 , \mathrm{x}=-2$$, and $$\mathrm{x}=-1$$.
Therefore, the possible eigenvalues of $$A$$ are $$1 , -2$$, and $$-1$$.

b) The invertibility of $$A$$ depends on whether the eigenvalues are nonzero. If all eigenvalues are nonzero, then the matrix is invertible. If some eigenvalues are zero, the matrix may not be invertible. To determine the invertibility of $$A$$, we would need more information, such as the rank and determinant of $$A$$.
by Diamond (64.2k points)

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