Suggested languages for you:

Americas

Europe

Problem 223

Find the characteristic polynomials and the eigenvalues of the matrices. (i) \(\begin{array}{rl}\mathrm{A}= & 2 & 3 \\ & 1 & 4\end{array} \mid\) (ii) $\mathrm{B}=\mid \begin{array}{rr}\cos \alpha & \sin \alpha \\ \mid-\sin \alpha & \cos \alpha \mid\end{array}$ (iii) $\begin{array}{rlr}\mathrm{C}= & 1 & 2 & 3 \mid \\ \mid 2 & 1 & 3 \mid \\\ \mid 3 & 3 & 6\end{array} \mid$

Expert verified

The short answer for the eigenvalues of the matrices are as follows:
- Matrix A: \(λ₁ = 1\), \(λ₂ = 5\)
- Matrix B: \(λ₁ = \cos\alpha + \sin\alpha\), \(λ₂ = \cos\alpha - \sin\alpha\)
- Matrix C: \(λ₁ = 2\), \(λ₂ = 3\), \(λ₃ = 3\)

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 9

If $\mathrm{F}(\mathrm{X})=\begin{array}{ccccc}1 & 0 & 1 \mid & \mid 2 & 1 & 0 \mid \\ & \mid 2 & 1 & 1 \mid & -\mid-1 & 1 & 1 \\ & 1 & 1 & 1 \mid & \mid 0 & 1 & 0\end{array} \mid$ $\begin{array}{rllll} & \mid 1 & 1 & 1 \mid & & \\ \mathrm{X}+ & \mid 1 & 0 & 1 \mid \mathrm{X}^{2}, & \text { and } \mathrm{B}= & 1 & 1 \\ & \mid 0 & 1 & 0 \mid & 1 & 0 \\ & \text { find } \mathrm{F}_{\mathrm{L}}(\mathrm{B}) & \text { and } \mathrm{F}_{\mathrm{R}}(\mathrm{B}) .\end{array}$

Chapter 9

Let \(\mathrm{T}\) be the linear operator on \(\mathrm{R}^{3}\) which is represented in the standard ordered basis by the matrix $\mathrm{A}=\mid \begin{array}{rrr}5 & -6 & -6 \\ -1 & 4 & 2 \\ 3 & -6 & -4\end{array}$ Find the minimal polynomial of \(\mathrm{T}\).

Chapter 9

Let \(\mathrm{V}\) be the vector space \(\mathrm{R}^{2}\) and let \(\mathrm{T}\) be the operator defined by $\quad \mathrm{T}(\mathrm{xy})=(2 \mathrm{x}-\mathrm{y}, \mathrm{x}+\mathrm{y})$ Let \(\mathrm{f}(\mathrm{x})=2+3 \mathrm{x}\) and \(\mathrm{g}(\mathrm{x})=\mathrm{x}+\mathrm{x}^{2} .\) Find \(\mathrm{f}(\mathrm{T})\) and \(\mathrm{g}(\mathrm{T})\)

Chapter 9

[A] Find the minimum polynomial \(\mathrm{m}(\lambda)\) of the matrix $\mathrm{A}=\begin{array}{cccc}2 & 1 & 0 & 0 \\ & 0 & 2 & 0 & 0 \mid \\ & 0 & 0 & 2 & 0 \\ & 0 & 0 & 0 & 5\end{array}$ [B] Let \(\mathrm{A}\) be a 3 by 3 matrix over the real field cannot be a zero of the polynomial, \(\varphi(\lambda)=\lambda^{2}+1\)

Chapter 9

Let \(\quad \varphi(\lambda)=-2-5 \lambda+3 \lambda^{2}\) $$ \begin{array}{rrrr}A= & \mid 1 & 2 \mid . & \text { Show that } & \varphi(A)=\mid 14 & 2 \mid \\ & \mid 3 & 1 \mid & & 13 & 14\end{array} $$

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner