Suggested languages for you:

Americas

Europe

Q38E

Expert-verifiedFound in: Page 324

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 $\lambda =2$

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.

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 $\lambda =2$

94% of StudySmarter users get better grades.

Sign up for free