Linear Algebra and Probability 3

Question 1. Consider the matrix

A =

  2 1 1−2 5 2 −1 1 4

 

(a) Find all eigenvalues of A. (b) For each eigenvalue of A, find the associated eigenvector(s).

Question 2. Consider the matrix

B =

 

1 2 3 4 −2 4 0 7 0 3 5 6 0 1 4 5

 

(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. Define a mapping F : V → U by matrix multiplication: 

 xy z

  7→

 

4 1 −2 2 −7 4 −1 8 −5 8 5 −6

    xy

z

 

(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 ?