What dimension or dimensions do you need to know to find the volume of a sphere?

The average number of acres burned by all wildfires in the united states is 780 acres with standard deviation 500 acres. of cour

shusha [124]

Mmmmmmmmmmmmmmmmmmmmmmmmmmmm

7 0

3 years ago

(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

HURRY PLS What is the solution to the system that is created by the equation y = 2 x + 10 and the graph shown below? On a coordi

marin [14]

Answer:

(-8, -6)

Step-by-step explanation:

When you graph the given equation on the given graph, you find the lines intersect at (-8, -6).

The solution to the system is (-8, -6).

__

The equation is given in slope-intercept form, so you can find the y-intercept (y=10) and draw a line with slope 2 from there. It will go through (-5, 0), so you know the solution lies in the 3rd quadrant and has an x-value less than -5. Only one answer choice matches.

5 0

3 years ago

Read 2 more answers

2.5d + 7.25 = 1 + 3.75d solve for d

dimulka [17.4K]

Answer:

5

Step-by-step explanation:

Subtract 1 from both sides. The equation becomes 2.5d + 6.25 = 3.75d

Subtract 2.5d from both sides. The equation becomes 6.25 = 1.25d

Divide 1.25 from both sides. You get 5 = d

4 0

3 years ago

Use this information for exercises 11-14.parents are older than their children.the ratio of a parent´sage to a child´s age chang

Shtirlitz [24]

I think 11 is yes, if the parent is 40 and the child is 20. (and they had kids when they were young) 12 is yes, if the parent is 30 and the child is 10. 13 is yes, if the parent is 60 and the child is 40. unlikely though, but who cares. 14 is no, the parent would have been really young when they gave birth to the child. like 10.

3 0

3 years ago