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Problem 242

Find the eigenvalues and the eigenvectors of the matrix \(\mathrm{A}\), where

Expert verified

The eigenvalues λ₁ and λ₂ are the solutions of the quadratic equation \(λ^2 - λ(a+d) + (ad - bc) = 0\). For each eigenvalue, find the corresponding eigenvectors by solving the system of linear equations \((A - λI)X = 0\), where A is the given matrix and I is the identity matrix. The eigenvectors can be obtained using row-reduction or any other preferred method.

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Chapter 10

Find the eigenvalues of \(\mathrm{A}\) and a basis for each eigenspace, where \(A=\mid \begin{array}{cc}2 & 21 \\ /-1 & 5 \mid\end{array}\)

Chapter 10

Find the eigenvalues of \(\mathrm{m}^{\prime} \mathrm{m}\) : (1) \(\quad \mathrm{m}=\begin{array}{rll}1 & 0 \\ & 10 & 0\end{array}\) (2) $\quad \mathrm{m}=\begin{array}{rr}\mid 0 & 1 \mid \\ & \mid-1 & 0\end{array}$ (3) \(\quad \mathrm{m}=\begin{array}{rl}11 & 1 \mid \\ & 10 & 11\end{array}\)

Chapter 10

Find the eigenvalues and a basis for each of the eigenspaces of the linear operator \(\mathrm{T}: \mathrm{P}_{2} \rightarrow \mathrm{P}_{2}\) defined by: $\mathrm{T}\left(\mathrm{a}+\mathrm{bx}+\mathrm{cx}^{2}\right)=(3 \mathrm{a}-2 \mathrm{~b})+(-2 \mathrm{a}+3 \mathrm{~b}) \mathrm{x}+(5 \mathrm{c}) \mathrm{x}^{2}$

Chapter 10

Reduce the matrix A of the linear transformation to a diagonal form where $\begin{array}{rrrr} & \mid 1 & 3 & 1 & 2 \\ \mathrm{~A}= & 10 & -1 & 1 & 3 \mid \\ 10 & 0 & 2 & 5 \mid \\ 10 & 0 & 0 & -21\end{array}$

Chapter 10

The characteristic values of the matrix $\begin{array}{rlrr} & /8 & 2 & -2 \mid \\ \mathrm{A}= & /3 & 3 & -1 \mid \\\ & /24 & 8 & -6\end{array}$

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