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Q38E

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Found in: Page 324

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# We are told that $\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]$ is an eigenvector of the matrix $\left[\begin{array}{ccc}4& 1& 1\\ -5& 0& -3\\ -1& -1& 2\end{array}\right]$ what is the associated eigenvalue?

Hence, the required eigenvalue is $\lambda =2$

See the step by step solution

## Step 1: Definition of the Eigenvectors

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

## Step 2: Finding eigenvalues

Consider $x=\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]$ be an eigen vector of the matrix $\left[\begin{array}{ccc}4& 1& 1\\ -5& 0& -3\\ -1& -1& 2\end{array}\right]$.

The objective is to find the associated eigenvalue.

If x is an eigen vector of A then $Ax=\lambda x$ where scalar $\lambda$ is called an eigenvalue of A.

Consider,

$Ax=\lambda x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}4& 1& 1\\ -5& 0& -3\\ -1& -1& 2\end{array}\right]\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]=\lambda \left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]\phantom{\rule{0ex}{0ex}}\left[\begin{array}{ccc}4& 1& 1\\ -5& 0& -3\\ -1& -1& 2\end{array}\right]=\lambda \left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]\phantom{\rule{0ex}{0ex}}\left[\begin{array}{c}2\\ -2\\ -2\end{array}\right]=\lambda \left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]\phantom{\rule{0ex}{0ex}}2\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]=\lambda \left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]$

Hence, the associated eigen value is $\lambda =2$