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Let $A$ be a $3×3$ matrix with $\operatorname{trace} (A) = 3$ and $\det (A) = 2$. If $1$ is an eigenvalue of $A$, then what are the eigenvalues of the matrix $A^2 - 2I$?

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How to handle the trace and det: the properties of Eigenvalues and Eigenvectors. – Frenzy Li Aug 12 '12 at 12:52
up vote 6 down vote accepted

Hint: Can you find the other eigenvalues of $A$ ? what do you know about the sum of all eigenvalues of $A$ ? what about their product ?

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No, Sir, I don't know, How to find other Eigen Values.I know that the sum of the eigenvalus is equal to trace and their multiplication is equal to the determinant. – ram Aug 12 '12 at 12:31
then if the eigenvalues are $a,b,1$ you have $a+b+1=3,ab=2$. can you solve for $a,b$ ? – Belgi Aug 12 '12 at 12:33
yes, Sir, a = 1 +- (i.root3)/2.... – ram Aug 12 '12 at 12:43
@ram, that isn't the answer. Here's a hint: $(x-a)(x-b)=x^2-(a+b)x+ab$ – J. M. Aug 12 '12 at 12:45

Once you have determined the eigenvalues of $A$, observe that they are distinct, hemce $A$ is diagonalizable (at least over $\mathbb C$), hence the eigenvalues of $A^2-2I$ are just the values $\lambda^2-2$ where $\lambda$ is an eigenvalue of $A$, that is $-1$, $a^2-2$, $b^2-2$ with the values found e.g. with Belgi's hint.

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