Linear Algebra and Probability 3

Please show all work. Thanks! See doc for ma. 

Question 1.

Consider the matrix

(a) Find all eigenvalues of A:

(b) For each eigenvalue of A; nd the associated eigenvector(s).

Question 2.

Consider the matrix

(a) Find the determinant of B:

(b) Is B invertible?

Question 3.

Let V be the set of 3  1 matrices and U be the set of 4  1 matrices. Dene a mapping F : V ! U by

matrix multiplication:

(a) Find the kernel of F:

(b) Find the image of F:

(c) Find the nullity of F:

(d) Find the rank of F:

(e) How are the nullity and the rank of F related to the dimensions of U and V ?