Find the eigen value of \(\begin{bmatrix}3 & 6 & -8\\ 0 & 0 & 6\\ 0 & 0 & 2\end{bmatrix}\)

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Let A = \(\begin{bmatrix}3 & 6 & -8\\ 0 & 0 & 6\\ 0 & 0 & 2\end{bmatrix}\)

The characteristic Equation is

|A – λI| = 0

Now,

A – λI = \(\begin{bmatrix}3 & 6 & -8\\ 0 & 0 & 6\\ 0 & 0 & 2\end{bmatrix}\) – λ \(\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}\)

= \(\begin{bmatrix}3 & 6 & -8\\ 0 & 0 & 6\\ 0 & 0 & 2\end{bmatrix}\) – \(\begin{bmatrix}λ & 0 & 0\\ 0 & λ & 0\\ 0 & 0 & λ\end{bmatrix}\)

= \(\begin{bmatrix}3-λ & 6 & -8\\ 0 & -λ & 6\\ 0 & 0 & 2-λ\end{bmatrix}\)

Now,

|A – λI| = 0

\(\begin{vmatrix}3-λ & 6 & -8\\ 0 & -λ & 6\\ 0 & 0 & 2-λ\end{vmatrix}\) = 0

(3-λ) {-λ . (2-λ) – 0} = 0

or, (3-λ) (-2λ + λ2) = 0

or, (3-λ) (λ2 – 2λ) = 0

or, 3λ2 – λ3 – 6λ + 2λ2 = 0

or, λ3 – 5λ2 + 6λ = 0

or λ (λ2 – 5λ + 6) = 0

or, λ = 0 and λ2 – 5λ + 6 = 0

Solving λ2 – 5λ + 6 =0 , we get,

λ = 2 and λ = 3

The Eigen values are 0, 2 and 3.

 

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