Suppose the matrix, $ A $, has eigenvectors \[\begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix},\begin{bmatrix} 2 \\ 1 \\ -2 \end{bmatrix},\begin{bmatrix} 5 \\ 0 \\ -3 \end{bmatrix}\]whose eigenvalues are 8, $-2$ and 4 respectively. Then, using the same order, $ A $ can be written in the form \[A = P \Lambda P^{-1}\]where\[P = \begin{bmatrix} \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \end{bmatrix}\]and\[\Lambda = \begin{bmatrix} \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \\ \boxed{} & \boxed{} & \boxed{} \end{bmatrix}\]

Given: The matrix A has eigenvectors 30-2, 21-2 and 50-3 whose eigenvalues are 8, -2 and 4 respectively. Then A can be written in the form A=PΛP-1 where P is an invertible matrix whose columns are the eigenvectors of A and Λ is a diagonal matrix whose main diagonal entries are the eigenvalues of A. Therefore, P=325010-2-2-3 and Λ=8000-20004

Answered: 3 2 Suppose the matrix, A, has…

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3 2 Suppose the matrix, A, has eigenvectors and whose eigenvalues are 8, – 2 and 4 respectively. Then, using the same order, A can be written in the form A = PAP-1 where P = and A =

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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Suppose the matrix, $ A $, has eigenvectors

\[
\begin{bmatrix}
3 \\
0 \\
-2
\end{bmatrix},
\begin{bmatrix}
2 \\
1 \\
-2
\end{bmatrix},
\begin{bmatrix}
5 \\
0 \\
-3
\end{bmatrix}
\]

whose eigenvalues are 8, $-2$ and 4 respectively. Then, using the same order, $ A $ can be written in the form

\[
A = P \Lambda P^{-1}
\]

where

\[
P =
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]

and

\[
\Lambda =
\begin{bmatrix}
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{} \\
\boxed{} & \boxed{} & \boxed{}
\end{bmatrix}
\]

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