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Expert-verified Found in: Page 323 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Is $\stackrel{\mathbf{⇀}}{\mathbf{v}}$an eigenvector of ${{\mathbf{A}}}^{{\mathbf{3}}}$? If so, what is the eigenvalue?

Yes, the required eigenvalue is ${\lambda }^{3}$ .

See the step by step solution

## Step 1: Definition of eigenvalue

An eigenvalue of A is a scalar ${\mathbf{\lambda }}$ such that the equation ${\mathrm{Av}}{=}{\mathrm{\lambda v}}$ has a nontrivial solution.

## Step 2: Find the eigenvalue

Assume that, A is an invertible matrix of order $n×n$, and $\stackrel{⇀}{v}$ is an eigenvector of A corresponding to eigenvalue $\lambda$ .

Is $\stackrel{⇀}{v}$ an eigenvector of ${A}^{3}$ or not, and find the eigenvalue of ${A}^{3}$.

If $\stackrel{⇀}{v}$ is an eigenvector of matrix A then,

$A\stackrel{⇀}{v}=\lambda \stackrel{⇀}{v}$

By the properties of eigenvalues and eigenvectors, if $\stackrel{⇀}{v}$ is an eigenvector of matrix A , then $\stackrel{⇀}{v}$ is an eigenvector of matrices ${A}^{2},{A}^{3},......{A}^{n}$, as shown below.

${A}^{2}\stackrel{⇀}{v}={\lambda }^{2}\stackrel{⇀}{v}\phantom{\rule{0ex}{0ex}}{A}^{3}\stackrel{⇀}{v}={\lambda }^{3}\stackrel{⇀}{v}\phantom{\rule{0ex}{0ex}}{A}^{n}\stackrel{⇀}{v}={\lambda }^{n}\stackrel{⇀}{v}$

Where, n is positive integer.

Now look at ${A}^{3}$obtained as,

${A}^{3}\stackrel{⇀}{v}=\lambda \stackrel{⇀}{v}$

Thus, $\stackrel{⇀}{v}$ is an eigenvector of ${A}^{3}$corresponding to eigenvalue ${\lambda }^{3}$.

Hence, $\stackrel{⇀}{v}$ is an eigenvector of ${A}^{3}$ , and the corresponding eigenvalue is ${\lambda }^{3}$ . ### Want to see more solutions like these? 