Find the eigenvalues of the matrix \(\begin{pmatrix}6 & 3 & -8\\ 0 & -2 & 0\\ 1 & 0 & -3\end{pmatrix}\)

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

The characteristics equation is

|A – λI| = 0

Now,

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

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

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

Now,

|A – λI| = 0

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

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

or, (6 – λ)(-2-λ)(-3-λ) + (0 – (16 + 8λ)) = 0

or, (6 – λ)(-2-λ)(-3-λ) + 8(-2 – λ) = 0

or, (-2 – λ) {(6 – λ)(-3-λ) + 8} = 0

or, (-2 – λ) {-18 – 6λ + 3λ + λ2 + 8} = 0

or, (-2 – λ) ( λ2 – 3λ – 10) = 0

Either,

-2-λ = 0

λ = 2

or,

λ2 – 3λ – 10 = 0

or, λ2 – (5 – 2)λ – 10 = 0

or, λ2 – 5λ + 2λ – 10 = 0

or, λ(λ – 5) + 2(λ – 5) = 0

or, (λ + 2) (λ – 5) = 0

or, λ = -2, 5

Therefore, Eigen values = -2, -2, 5

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