### Consider the following.\[ A = \begin{bmatrix}1 & 0 & 2 \\3 & -1 & 3 \\2 & 0 & 1\end{bmatrix}\](a) Compute the characteristic polynomial of \( A \).(b) Compute the eigenvalues and bases of the corresponding eigenspaces of \( A \).(c) Compute the algebraic and geometric multiplicity of each eigenvalue. Recall that the general method to find the eigenvalues of a matrix \( A \) is to find the solutions \( \lambda \) of the equation \( \text{det}(A - \lambda I) = 0 \). This is because \( \lambda \) is an eigenvalue of \( A \) if and only if \( A - \lambda I \) is noninvertible, which is the case exactly when \( \text{det}(A - \lambda I) = 0 \).Notice that treating \( \lambda \) as an unknown variable, the determinant \( \text{det}(A - \lambda I) \) is an expression rather than a number and is called the characteristic polynomial.First, set up the characteristic polynomial.\[\text{det}(A - \lambda I) = \text{det} \left( \begin{bmatrix}1 & 0 & 2 \\3 & -1 & 3 \\2 & 0 & 1 \end{bmatrix}- \begin{bmatrix}\lambda & 0 & 0 \\0 & \lambda & 0 \\0 & 0 & \lambda\end{bmatrix}\right) \]\[= \begin{vmatrix}1 - \lambda & 0 & 2 \\3 & -1 - \lambda & 3 \\2 & 0 & 1 - \lambda\end{vmatrix}\]To find the determinant, expand along the first row and substitute the appropriate expressions from the matrix.\[\text{det}(A - \lambda I) = a_{11} \text{det}(A_{11}) - a_{12} \text{det}(A_{12}) + a_{13} \text{det}(A_{13})\]\[= (\underline{\hspace{0.5cm}}) \text{det}(A_{11}) - (0) \text{det}(A_{12}) + (\underline{\hspace{0.5cm}}) \text{det}(A_{13})\]Now, find the required submatrices that have nonzero coefficients in the above equation.\[\text{det}(A_{11}) = (-1 - \lambda) \left( \begin{vmatrix} \underline

Consider the following. 0 2 A = 3 -1 3 20 1. (a) Compute the characteristic polynomial of A. (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (c) Compute the algebraic and geometric multiplicity of each eigenvalue.

Consider the following. 0 2 A = 3 -1 3 20 1. (a) Compute the characteristic polynomial of A. (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (c) Compute the algebraic and geometric multiplicity of each eigenvalue.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Consider the following.

\[ A =
\begin{bmatrix}
1 & 0 & 2 \\
3 & -1 & 3 \\
2 & 0 & 1
\end{bmatrix}
\]

(a) Compute the characteristic polynomial of \( A \).

(b) Compute the eigenvalues and bases of the corresponding eigenspaces of \( A \).

(c) Compute the algebraic and geometric multiplicity of each eigenvalue.

Recall that the general method to find the eigenvalues of a matrix \( A \) is to find the solutions \( \lambda \) of the equation \( \text{det}(A - \lambda I) = 0 \). This is because \( \lambda \) is an eigenvalue of \( A \) if and only if \( A - \lambda I \) is noninvertible, which is the case exactly when \( \text{det}(A - \lambda I) = 0 \).

Notice that treating \( \lambda \) as an unknown variable, the determinant \( \text{det}(A - \lambda I) \) is an expression rather than a number and is called the characteristic polynomial.

First, set up the characteristic polynomial.

\[
\text{det}(A - \lambda I) = \text{det} \left(
\begin{bmatrix}
1 & 0 & 2 \\
3 & -1 & 3 \\
2 & 0 & 1
\end{bmatrix}
-
\begin{bmatrix}
\lambda & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda
\end{bmatrix}
\right)
\]

\[
=
\begin{vmatrix}
1 - \lambda & 0 & 2 \\
3 & -1 - \lambda & 3 \\
2 & 0 & 1 - \lambda
\end{vmatrix}
\]

To find the determinant, expand along the first row and substitute the appropriate expressions from the matrix.

\[
\text{det}(A - \lambda I) = a_{11} \text{det}(A_{11}) - a_{12} \text{det}(A_{12}) + a_{13} \text{det}(A_{13})
\]

\[
= (\underline{\hspace{0.5cm}}) \text{det}(A_{11}) - (0) \text{det}(A_{12}) + (\underline{\hspace{0.5cm}}) \text{det}(A_{13})
\]

Now, find the required submatrices that have nonzero coefficients in the above equation.

\[
\text{det}(A_{11}) = (-1 - \lambda) \left( \begin{vmatrix} \underline

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