Question 1

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Accepted Answer

C

Question 2

Solve the initial value problem.&#10;

Accepted Answer

A

Question 3

For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.&#10;

Accepted Answer

C

Question 4

The characteristic polynomial of a 5 &#215; 5 matrix is given below. Find the eigenvalues and their multiplicities.&#10;

Accepted Answer

The characteristic polynomial is missing its constant term, which implies it is divisible by $\lambda^2$, indicating $\lambda = 0$ has multiplicity 2. Factoring the polynomial as $\lambda^2(\lambda^3 - 24\lambda^2 + 189\lambda - 486)$ and solving for the roots of the cubic part will give the other eigenvalues. The correct factorization of the cubic part gives $\lambda = 9$ (multiplicity 2) and $\lambda = 6$ (multiplicity 1), making option A correct.

Question 5

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that &#10;  &#10;

Accepted Answer

The answer of Diagonalize the matrix A, if possible. That...

Question 6

Find the eigenvalues of the given matrix.&#10;  )6

Accepted Answer

The characteristic equation of the matrix is:$$\det(A - \lambda I) = 0$$$$(56 - \lambda)(-49 - \lambda) - (10)(-275) = 0$$$$\lambda^2 - 5\lambda - 1260 = 0$$$$(\lambda - 6)(\lambda + 210) = 0$$Therefore, the eigenvalues are $\lambda = 6$ and $\lambda = -210$.

Question 7

Find the characteristic equation of the given matrix.&#10;

Accepted Answer

The characteristic equation of a matrix is found by taking the determinant of $A - \lambda I$, where $I$ is the identity matrix and $\lambda$ is a scalar. For a triangular matrix (including diagonal matrices), the determinant is the product of the diagonal elements minus $\lambda$. Thus, the characteristic equation of the given matrix is $(9-\lambda)^2(7-\lambda)(5-\lambda)=0$, as the diagonal elements are 9, 7, 5, and 9.

Question 8

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Accepted Answer

The matrix of the linear transformation $T$ relative to $B$ and $C$ is$$\left[\begin{array}{rr}5 & 5 \6 & 12 \-5 & 7\end{array}\right]$$because$$\begin{aligned}\mathrm { T } \left( \mathbf { b } _ { 1 } \right) &=5 c _ { 1 } + 6 c _ { 2 } - 5 c _ { 3 } \&=5\left[ \begin{array} { l } 1 \0 \0\end{array} \right] + 6\left[ \begin{array} { l } 0 \1 \0\end{array} \right] - 5\left[ \begin{array} { l } 0 \0 \1\end{array} \right] \&=\left[ \begin{array} { r } 5 \6 \-5\end{array} \right] \\mathrm { T } \left( \mathbf { b } _ { 2 } \right) &=5 c _ { 1 } + 12 c _ { 2 } + 7 c _ { 3 } \&=5\left[ \begin{array} { l } 1 \0 \0\end{array} \right] + 12\left[ \begin{array} { l } 0 \1 \0\end{array} \right] + 7\left[ \begin{array} { l } 0 \0 \1\end{array} \right] \&=\left[ \begin{array} { r } 5 \12 \7\end{array} \right]\end{aligned}$$

Question 9

Find the characteristic equation of the given matrix.&#10;

Accepted Answer

The characteristic equation of a matrix $A$ is found by calculating the determinant of $A - \lambda I$, where $I$ is the identity matrix and $\lambda$ are the eigenvalues. For a triangular matrix (including diagonal matrices), the eigenvalues are the entries on the main diagonal. Thus, the characteristic equation is $(8 - \lambda)(3 - \lambda)(-6 - \lambda)(-5 - \lambda) = 0$.

Question 10

The characteristic polynomial of a 5 &#215; 5 matrix is given below. Find the eigenvalues and their multiplicities.&#10;

Accepted Answer

The answer of The characteristic polynomial of a 5 &#215;...

Question 11

For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.&#10;

Accepted Answer

The answer of For the given matrix A, find a...

Question 12

The characteristic polynomial of a 5 &#215; 5 matrix is given below. Find the eigenvalues and their multiplicities.&#10;

Accepted Answer

The answer of The characteristic polynomial of a 5 &#215;...

Question 13

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that &#10;  &#10;

Accepted Answer

The answer of Diagonalize the matrix A, if possible. That...

Question 14

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that &#10;  &#10;

Accepted Answer

The answer of Diagonalize the matrix A, if possible. That...

Question 15

Use the inverse power method to determine the smallest eigenvalue of the matrix A.&#10;Assume that the eigenvalues are roughly 1.3, 2.1, and 15.

Accepted Answer

The answer of Use the inverse power method to determine...

Question 16

Find the eigenvalues of the given matrix.&#10;

Accepted Answer

The answer of Find the eigenvalues of the given matrix.&#10;...

Question 17

For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.&#10;

Accepted Answer

The answer of For the given matrix and eigenvalue, find...

Question 18

For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.&#10;

Accepted Answer

The answer of For the given matrix and eigenvalue, find...

Question 19

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that &#10;  &#10;

Accepted Answer

The answer of Diagonalize the matrix A, if possible. That...

Question 20

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Accepted Answer

The answer of 11eb3ac5_a8b1_925d_9b48_0fb7571fe2ce_TB7504_0011eb3ac5_a8b1_925f_9b48_ebb22f452469_TB7504_00...

Deck 5: Eigenvalues and Eigenvectors