Eigenvalues of a Matrix and its Inverse Consider the following matrix -12 30 -5 13 (a) Determine the characteristic polynomial of A and its eigenvalues. (b) Determine A-1. (c) Determine the characteristic polynomial of A-' and its eigenvalues. (d) Is there any relation between the eigenvalues of A and those of A-1?

Eigenvalues of a Matrix and its Inverse Consider the following matrix -12 30 -5 13 (a) Determine the characteristic polynomial of A and its eigenvalues. (b) Determine A-1. (c) Determine the characteristic polynomial of A-' and its eigenvalues. (d) Is there any relation between the eigenvalues of A and those of A-1?

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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Question

# 3. Eigenvalues of a Matrix and its Inverse

Consider the following matrix:

\[
A = \begin{bmatrix} -12 & 30 \\ -5 & 13 \end{bmatrix}
\]

(a) Determine the characteristic polynomial of $ A $ and its eigenvalues.

(b) Determine $ A^{-1} $.

(c) Determine the characteristic polynomial of $ A^{-1} $ and its eigenvalues.

(d) Is there any relation between the eigenvalues of $ A $ and those of $ A^{-1} $?

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