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
Heather wants to buy new carpet for her living room floor. The floor is in the shape of a rectangle. Its length is 19 feet and i
SCORPION-xisa [38]

Answer:

$2431

Step-by-step explanation:

wish u was Heather lol

┈┈┈┈▅┈┈┈┈┈▅ ┈┈

┈┈┈▕┈┈┈╱╲▕▀┈┈┈

┈┈┈╱╲┈┈▏▕╱╲┈┈┈

┈┈┈▏▕╱╲▏▎▏▕╱╲┈▃

┈╱╲▏▎▅▂▅▂▏▎▏▎▏▏

▂▏▎▏▕╭┳┳╮▏┊▏▕╱╲

▏▏┊▏▎┃┊┊┃▏▎▏▎▏▕

▇▆▅▃▂┻┻┻┻▂▃▅▆▇▉

6 0
3 years ago
How many seconds are in 45 minutes?
fgiga [73]
45 minutes equal to 2700 seconds.
8 0
3 years ago
Read 2 more answers
I need help on this guys pls help me
STatiana [176]

Answer:

The answer is A

Step-by-step explanation:

The reason A is correct is because all of the other transformations change the size of the triangle, and A is the only answer that only reflects the image over the y axis without changing it, thus making it similar to the preimage

4 0
2 years ago
Solve for x! Please try and show work. ANSWER ASAP SCAM ANSWERS WILL BE REPORTED. Also will award brainliest to CORRECT answer
Tamiku [17]

Answer:

x = 12 degrees

Step-by-step explanation:

m < LNM = (70+136)/2

= 206 / 2

= 103 degrees

So 7x + 19 = 103

7x = 103 - 19 = 84

x = 84/7

x = 12.

8 0
3 years ago
Please help ! This is 7th grade math and I’ve been stuck on this question for 30 minutes!
FinnZ [79.3K]

\dfrac{1}{22}\:\dfrac{\text{in}}{\text{hr}}

Step-by-step explanation:

The amount of time from 10pm to 7am is 9 hours. However, the time indicated is 7:10am. Since there are 60 minutes in an hour, 10 min is equal to 1/6 of an hour. Therefore, the total time that the water level of the pool decreased is 9\frac{1}{6} hours or \frac{55}{6} hours. To find the rate at which the water level decreased, we divide the change in the water (in inches) by the time it took to reach that level. So we write

rate of change = \dfrac{\text{change\:in\:height}}{\text{change\:in\:time}}

\:\:\:\:\:\:\:= \dfrac{\frac{5}{12}\:\text{in}}{\frac{55}{6}\:\text{hrs}}

\:\:\:\:\:\:\:=\dfrac{5}{12}\cdot\dfrac{6}{55}\:\dfrac{\text{in}}{\text{hr}}

\:\:\:\:\:\:\:= \frac{1}{22}\:\dfrac{\text{in}}{\text{hr}}

7 0
3 years ago
Other questions:
  • Let z= -5 sqrt 3/2 + 5/2i and w=1 + sqrt 3i
    7·1 answer
  • Please can you help me with questions 1-6
    11·2 answers
  • Mala is y years old her sister is 4 years older than mala write an expression in simplest form that repersents the sum of the ag
    6·1 answer
  • A test statistic is found to have a P-value of .015. Which of the following statements are true?
    5·1 answer
  • Factor xy + 3x - 2x^2 completely​
    10·1 answer
  • Rewrite 0.6+(-1)-(-0.8)-0.4=0
    6·1 answer
  • Y'all- this question make's 0 sense to my big brain help me out
    13·1 answer
  • Does Someone Know The Answer To This ?
    12·1 answer
  • What is the quotient of 22/3
    7·1 answer
  • Answer as soon as possible pls
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!