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(1) (10 points) Find the characteristic polynomial of A (2) (5 points) Find all eigenvalues of A. You are allowed to use your ca

Yuri [45]

Answer:

Step-by-step explanation:

Since this question is lacking the matrix A, we will solve the question with the matrix

$\left[\begin{matrix}4 & -2 \\ 1 & 1 \end{matrix}\right]$

so we can illustrate how to solve the problem step by step.

a) The characteristic polynomial is defined by the equation $det(A-\lambdaI)=0$ where I is the identity matrix of appropiate size and lambda is a variable to be solved. In our case,

$\left|\left[\begin{matrix}4-\lamda & -2 \\ 1 & 1-\lambda \end{matrix}\right]\right|= 0 = (4-\lambda)(1-\lambda)+2 = \lambda^2-5\lambda+4+2 = \lambda^2-5\lambda+6$

So the characteristic polynomial is $\lambda^2-5\lambda+6=0$ .

b) The eigenvalues of the matrix are the roots of the characteristic polynomial. Note that

$\lambda^2-5\lambda+6=(\lambda-3)(\lambda-2) =0$

So $\lambda=3, \lambda=2$

c) To find the bases of each eigenspace, we replace the value of lambda and solve the homogeneus system(equalized to zero) of the resultant matrix. We will illustrate the process with one eigen value and the other one is left as an exercise.

If $\lambda=3$ we get the following matrix

$\left[\begin{matrix}1 & -2 \\ 1 & -2 \end{matrix}\right]$ .

Since both rows are equal, we have the equation

$x-2y=0$ . Thus x=2y. In this case, we get to choose y freely, so let's take y=1. Then x=2. So, the eigenvector that is a base for the eigenspace associated to the eigenvalue 3 is the vector (2,1)

For the case $\lambda=2$ , using the same process, we get the vector (1,1).

d) By definition, to diagonalize a matrix A is to find a diagonal matrix D and a matrix P such that $A=PDP^{-1}$ . We can construct matrix D and P by choosing the eigenvalues as the diagonal of matrix D. So, if we pick the eigen value 3 in the first column of D, we must put the correspondent eigenvector (2,1) in the first column of P. In this case, the matrices that we get are

$P=\left[\begin{matrix}2&1 \\ 1 & 1 \end{matrix}\right], D=\left[\begin{matrix}3&0 \\ 0 & 2 \end{matrix}\right]$

This matrices are not unique, since they depend on the order in which we arrange the eigenvalues in the matrix D. Another pair or matrices that diagonalize A is

$P=\left[\begin{matrix}1&2 \\ 1 & 1 \end{matrix}\right], D=\left[\begin{matrix}2&0 \\ 0 & 3 \end{matrix}\right]$

which is obtained by interchanging the eigenvalues on the diagonal and their respective eigenvectors

4 0

3 years ago

When is making a strawberry cake for her brother's birthday the cake Pan is right rectangular prism 20 cm wide by 28 cm long whe

Anna [14]

Answer:

3 2/7 or 23/7 cms

Step-by-step explanation:

So we know the volume of a prism = L * W * H

And with the given info we know = L=28 and W=20 and Vol=1840

we solve for H and we get 23/7

8 0

3 years ago

Two teams were working together to complete an Escape Room for Ms. Bradshaw. The players scores are totaled to get their finish

11Alexandr11 [23.1K]

Answer:

Matt's team

Step-by-step explanation:

Sarah's team finished in 1 hour and 30 min

Matt's team finished in 1 hour and 15 min

3 0

3 years ago

Susan’s elevation at the top of a mountain is 1530 meters.After four hours of climbing down her elevation is 680 meters. What is

VARVARA [1.3K]

Answer:

212.5/h

Step-by-step explanation:

subtract the new value from the old value, then divide by the time.

(1530 - 680)/4

850/4

212.5/1h

8 0

3 years ago

Read 2 more answers

A box of pancake mix containing 2 1/4 pounds costs $3.60. What is the cost per pound?

mote1985 [20]

Answer:

$1.60

Step-by-step explanation:

$3.60 divided by 2 1/4 is $1.60

6 0

3 years ago