a. Let A be a 2 by 2 matrix. Show that the characteristic polynomial p(x) is p(\) = \² — tr(A)λ + det(A) where tr(A) is the trace of the matrix A. b. If tr(A) = algebraic multiplicity of A? = 2a and det(A) = a², what are the eigenvalues of A and what is the
a. Let A be a 2 by 2 matrix. Show that the characteristic polynomial p(x) is p(\) = \² — tr(A)λ + det(A) where tr(A) is the trace of the matrix A. b. If tr(A) = algebraic multiplicity of A? = 2a and det(A) = a², what are the eigenvalues of A and what is the
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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![**Matrix Characteristic Polynomial and Eigenvalues**
Let \( \mathbf{A} \) be a 2 by 2 matrix.
**a.** Show that the characteristic polynomial \( p(\lambda) \) is
\[
p(\lambda) = \lambda^2 - \text{tr}(\mathbf{A})\lambda + \det(\mathbf{A})
\]
where \( \text{tr}(\mathbf{A}) \) is the trace of the matrix \( \mathbf{A} \).
**b.** If \( \text{tr}(\mathbf{A}) = 2a \) and \( \det(\mathbf{A}) = a^2 \), what are the eigenvalues of \( \mathbf{A} \) and what is the algebraic multiplicity of \( \mathbf{A} \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F039207b8-632a-4dc4-a3c4-fd77b3c684b4%2F1ff07ab3-e0f1-4bad-a9e7-56753663f11b%2F35nr43_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Characteristic Polynomial and Eigenvalues**
Let \( \mathbf{A} \) be a 2 by 2 matrix.
**a.** Show that the characteristic polynomial \( p(\lambda) \) is
\[
p(\lambda) = \lambda^2 - \text{tr}(\mathbf{A})\lambda + \det(\mathbf{A})
\]
where \( \text{tr}(\mathbf{A}) \) is the trace of the matrix \( \mathbf{A} \).
**b.** If \( \text{tr}(\mathbf{A}) = 2a \) and \( \det(\mathbf{A}) = a^2 \), what are the eigenvalues of \( \mathbf{A} \) and what is the algebraic multiplicity of \( \mathbf{A} \)?
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