Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 04 (c)

MATH152 April 2013

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Question B 04 (c)
Let A be the $3\times 3$ matrix given by ${\begin{bmatrix}-7&0&1\\0&3&0\\-2&0&-4\end{bmatrix}}$ It is known that $\lambda _{1}=3$ is an eigenvalue of A. Find the other eigenvectors associated with $\lambda _{2}$ and $\lambda _{3}$

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Hint
Compute the matrices $\displaystyle A-\lambda I$ and then check eigenvectors by inspecting the nullspace of this new matrix, for each value of $\lambda$. It will simplify matter when you use vectors with the middle entry equal to 0.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We compute the matrices $\displaystyle A-\lambda I$ and check its nullspace ${\begin{bmatrix}-7&0&1\\0&3&0\\-2&0&-4\end{bmatrix}}-{\begin{bmatrix}-5&0&0\\0&-5&0\\0&0&-5\end{bmatrix}}={\begin{bmatrix}-2&0&1\\0&8&0\\-2&0&1\end{bmatrix}}$ and from this we can see that ${\begin{bmatrix}1\\0\\2\end{bmatrix}}$ is an eigenvector associated to $\lambda =-5$ (to get all eigenvectors, multiply this vector by any non-zero number). Repeating this with $\displaystyle \lambda =-6$ ${\begin{bmatrix}-7&0&1\\0&3&0\\-2&0&-4\end{bmatrix}}-{\begin{bmatrix}-6&0&0\\0&-6&0\\0&0&-6\end{bmatrix}}={\begin{bmatrix}-1&0&1\\0&9&0\\-2&0&2\end{bmatrix}}$ and from this we can see that ${\begin{bmatrix}1\\0\\1\end{bmatrix}}$ is an eigenvector associated to $\lambda =-6$ (to get all eigenvectors, multiply this vector by any non-zero number).

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