Question
Find eigenvalues of matrix:
$$ A = \left[ \begin{matrix}5&2&0\\2&5&0\\-2&\frac{ 6 }{ 5 }&\frac{ 3 }{ 2 }\end{matrix} \right] $$Answer
The eigenvalues of matrix A are:
$$ \lambda_1 = 3 ~ \text{ , } ~ \lambda_2 = \dfrac{ 3 }{ 2 } ~ \text{ and } ~ \lambda_3 = 7 $$Explanation
Step 1 : The characteristic polynomial for matrix A is:
$$ p(\lambda) = -\lambda^3+\frac{ 23 }{ 2 }\lambda^2-36\lambda+\frac{ 63 }{ 2 } $$
Step 2 : Eigenvalues are roots of the characteristic polynomial, so the equation
$$ -\lambda^3+\frac{ 23 }{ 2 }\lambda^2-36\lambda+\frac{ 63 }{ 2 } = 0 $$yields the above eigenvalues.
( click here to view an explanation on how to solve this equation.)
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