1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
sweet [91]
3 years ago
15

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

lculator or computer to help you factor the characteristic polynomial. (3) (15 points) Find a basis for each eigenspace of A. (4) (5 points) Find a matrix P that diagonalizes A. Is this matrix P unique? If not, give a different P that also diagonalizes A together with the matrix D for this new P
Mathematics
1 answer:
Yuri [45]3 years ago
4 0

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

You might be interested in
A train travels at a speed of 48.6 km/h. Express this speed in<br>(ii) cm/min.<br>(1) m/s,<br>​
KATRIN_1 [288]

Answer:

Step-by-step explanation:

Conversion factors:

1 km = 1000 m

1 m = 100 cm

1 h = 60 min

1 min = 60 sec

48.6 km/h = 81,000 cm/min = 13.5 m/sec

4 0
3 years ago
There are 72 girls and 60 boys waiting to see A play in the school auditorium. They are seated in rows in the auditorium with th
borishaifa [10]
To answer this problem you first find the GCF (Greatest common factor) between 60 and 72, which is 12. Next, to find how many rows of girls there are you divide 72 by 12. You get 6. There are 6 rows of girls. Next, you divide the number of boys which is 60 by twelve to find how many rows of boys there are and you get 5. there are 5 rows of boys.
5 0
3 years ago
Read 2 more answers
PLEASE HELP!!<br> It’s asking for the measure of Arc KL in degrees
barxatty [35]

Answer:

30 degrees

Step-by-step explanation:

5 0
3 years ago
What is the domain of the function f(x)= 1/3x+2
frutty [35]
If it is 1/(3x+2), 3x+2≠0 =>x≠-2/3, the domain is all real number except -2/3
if it is 1/(3x)+2, 3x≠0, x≠0, the domain is all real number except 0

It is hard to tell from your equation what the demonstrator is.
3 0
3 years ago
Read 2 more answers
A flower arrangement was marked down 12.5%. if the flowers originally cost $20, what is the amount of the discount?
WINSTONCH [101]
The correct answer would be B :)
7 0
3 years ago
Read 2 more answers
Other questions:
  • Suppose that the terminal point determined by t is the point 12 13 , 5 13 on the unit circle. Find the terminal point determined
    5·1 answer
  • What's 7.2j = $93.24
    10·1 answer
  • Owen solves the equation below and his steps are shown. When he plugged his solution back into the original equation, he did not
    8·1 answer
  • Vroom Vacuums sells the Tornado vacuum cleaner. Each Tornado has a one-year warranty that covers any product defects. When custo
    7·1 answer
  • A Recipe for party mix calls for 3/4 cup of cereal , 1/4 Cup of walnuts, 5/8 cup of crackers , and 1/2 cup of raisins . determin
    15·2 answers
  • 4h + 7 = -8 – h .......
    15·1 answer
  • Calculate: [-22(-0.9)] – [(-3) x 2.3]
    11·2 answers
  • State the quadrant or axis that each point lies in. (-5,2)
    14·1 answer
  • Khan academy What is the slope of the line?
    5·1 answer
  • Alicia can walk 1/4 mile every 1/5 hour. At this rate, how far can Alicia walk in one hour?
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!