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Answer:
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)