Question
Find eigenvalues of matrix:
$$ A = \left[ \begin{matrix}-2&5&4\\5&7&5\\4&5&-2\end{matrix} \right] $$Answer
The eigenvalues of matrix A are:
$$ \lambda_1 = -3 ~ \text{ , } ~ \lambda_2 = -6 ~ \text{ and } ~ \lambda_3 = 12 $$Explanation
Step 1 : The characteristic polynomial for matrix A is:
$$ p(\lambda) = -\lambda^3+3\lambda^2+90\lambda+216 $$
Step 2 : Eigenvalues are roots of the characteristic polynomial, so the equation
$$ -\lambda^3+3\lambda^2+90\lambda+216 = 0 $$yields the above eigenvalues.
( click here to view an explanation on how to solve this equation.)
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