A = 4 0 0 1 3 0 −2 3 −1 Find the characteristic polynomial for the matrix A. (Write your answer in terms of λ.) Find the real eigenvalues for the matrix A. (Enter your answers as a comma-separated list.) λ = Find a basis for each eigenspace for the matrix A.

Step-by-step explanation:

We are given the matrix

a) To find the characteristic polynomial we calculate where I is the identity matrix of appropiate size. in this case the characteristic polynomial is

Since this matrix is upper triangular, its determinant is the multiplication of the diagonal entries, that is

which is the characteristic polynomial of A.

b) To find the eigenvalues of A, we find the roots of the characteristic polynomials. In this case they are

c) To find the base associated to the eigenvalue lambda, we replace the value of lambda in the expression and solve the system by finding a base for its solution space. We will show this process for one value of lambda and give the solution for the other cases.

Consider . We get the matrix

The second line gives us the equation x-y =0. Which implies that x=y. The third line gives us the equation -2x+3y-5z=0. Since x=y, it becomes y-5z =0. This implies that y = 5z. So, combining this equations, the solution of the homogeneus system is given by

So, the base for this eigenspace is the vector (5,5,1).

If then the base is (0,4,3) and if then the base is (0,0,1)

soo this can be solve by using the formula:

f = p( 1 + i)^ n

where f is the future profit

p is the initial profit

i is the percent increase

n is the number of years

a . f = 30000 ( 1.05)^n

b. f = 30000 ( 1.05)^15

f = $62,368