Find the eigenvalues and eigenvectors of A geometrically over the real numbers ℝ. (If an eigenvalue does not exist, enter DNE. If an eigenvector does not exist, enter DNE in any single blank.) A = 0 1 1 0 (reflection in the line y = x) λ1 = has eigenspace span (smaller λ-value) λ2 = has eigenspace span (larger λ-value)

We want to find the eigenvalues and eigenvectors of the given matrix.

The eigenvalues are 1 and -1 and the correspondent eigenvectors are:

 and  

We have the matrix:

To find the eigenvalues λ, we need to solve:

This gives:

Then the eigenvalues are 1 and -1

Now let's get the eigenvectors.

For λ = 1 we must solve:

Then a = -b

This eigenvector is:

(the factor is for normalization)

We do the same thing for the other eigenvalue:

Now we have a = b

If you want to learn more, you can read:

link

∴The eigen values are -1,1.

The eigen vector for 1 is     .

The eigen vector for = - 1 is     .

Step-by-step explanation:

Given matrix is

To find the eigen values of the given matrix, we set

∴The eigen values are -1,1.

For

Let the eigen vector for is

∴

let

The eigen vector for 1 is     .

For

Let the eigen vector for is

∴

and

From the above equations, we get

Let

Then,

The eigen vector for = - 1 is     .

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

hope this !

answer: 8

step-by-step explanation: if you add the possibilities 5 different flavor and 3 different sizes 5+3=8