3. This matrix has repeated eigenvalues for which value(s) of a? Show your wor -1] a M 4
3. This matrix has repeated eigenvalues for which value(s) of a? Show your wor -1] a M 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**3. This matrix has repeated eigenvalues for which value(s) of \(a\)? Show your work.**
Given the matrix \(M\):
\[
M = \begin{bmatrix}
a & -1 \\
4 & 4
\end{bmatrix}
\]
(Show the matrix \(M\) above your solution.)
---
To find the value(s) of \(a\) for which the matrix \(M\) has repeated eigenvalues, let's follow these steps:
1. **Write the characteristic equation of the matrix \(M\).**
The characteristic equation for a matrix \(M\) is given by the determinant of \((M - \lambda I) = 0\), where \(\lambda\) is an eigenvalue and \(I\) is the identity matrix:
\[
\text{det} \left( \begin{bmatrix}
a & -1 \\
4 & 4
\end{bmatrix} - \lambda \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} \right) = 0
\]
2. **Subtract \(\lambda\) times the identity matrix from matrix \(M\):**
\[
M - \lambda I = \begin{bmatrix}
a & -1 \\
4 & 4
\end{bmatrix} - \begin{bmatrix}
\lambda & 0 \\
0 & \lambda
\end{bmatrix} = \begin{bmatrix}
a - \lambda & -1 \\
4 & 4 - \lambda
\end{bmatrix}
\]
3. **Calculate the determinant of the resulting matrix:**
\[
\text{det} \left( \begin{bmatrix}
a - \lambda & -1 \\
4 & 4 - \lambda
\end{bmatrix} \right) = (a - \lambda)(4 - \lambda) - (-1 \cdot 4)
\]
4. **Simplify the equation:**
\[
(a - \lambda)(4 - \lambda) + 4 = a(4 - \lambda) - \lambda (4 - \lambda) + 4
\]
\[
= 4a - a\lambda - 4\lambda + \lambda^2 + 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d2fdd51-a813-4b36-89e9-f9581acfc2ee%2F8866b45e-793a-469e-aefe-548409200284%2Fviqigee_processed.png&w=3840&q=75)
Transcribed Image Text:**3. This matrix has repeated eigenvalues for which value(s) of \(a\)? Show your work.**
Given the matrix \(M\):
\[
M = \begin{bmatrix}
a & -1 \\
4 & 4
\end{bmatrix}
\]
(Show the matrix \(M\) above your solution.)
---
To find the value(s) of \(a\) for which the matrix \(M\) has repeated eigenvalues, let's follow these steps:
1. **Write the characteristic equation of the matrix \(M\).**
The characteristic equation for a matrix \(M\) is given by the determinant of \((M - \lambda I) = 0\), where \(\lambda\) is an eigenvalue and \(I\) is the identity matrix:
\[
\text{det} \left( \begin{bmatrix}
a & -1 \\
4 & 4
\end{bmatrix} - \lambda \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} \right) = 0
\]
2. **Subtract \(\lambda\) times the identity matrix from matrix \(M\):**
\[
M - \lambda I = \begin{bmatrix}
a & -1 \\
4 & 4
\end{bmatrix} - \begin{bmatrix}
\lambda & 0 \\
0 & \lambda
\end{bmatrix} = \begin{bmatrix}
a - \lambda & -1 \\
4 & 4 - \lambda
\end{bmatrix}
\]
3. **Calculate the determinant of the resulting matrix:**
\[
\text{det} \left( \begin{bmatrix}
a - \lambda & -1 \\
4 & 4 - \lambda
\end{bmatrix} \right) = (a - \lambda)(4 - \lambda) - (-1 \cdot 4)
\]
4. **Simplify the equation:**
\[
(a - \lambda)(4 - \lambda) + 4 = a(4 - \lambda) - \lambda (4 - \lambda) + 4
\]
\[
= 4a - a\lambda - 4\lambda + \lambda^2 + 4
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