What is as characteristic polynomial what are as

Question

1. Which of the following maps T : R³ → R² are linear transformations? Justify your answers.

2. a) Explain how elementary row operations can be used to find the inverse of a matrix (if it has one).

b) Let A be the 5 x 5 square matrix

Using the method you outlined in (a), either find A's inverse, or else demonstrate that A has no inverse.

3. Let

Is a invertible? Justify your answer.

4. Let

a) Write down A^T, the transpose of A.

b) Calculate |A| and |A^T|.

c) Calculate AA^T.

d) Calculate |AA^T|.

5. Let

a) Show how to reduce A to reduced echelon form using elementary row operations.

b) Find the general solution to the system of linear equations

v + 2w + 3x + y + 2z = 3

        3w + 4x + y + z = 0

                         v+ z = 1

Justify your answer.

6. Let x be a real number, and let

a) Calculate |A|. Show your wonting.

b) For which value(s) of x, if any, is A not invertible? Justify your answer.

7. Suppose r is a real number, and consider the system of equations

 x + 2y = 3r

 2x - ry = 1

Rx - 2ry = r

a) For which values of r (if any) does this system of equations have exactly one solution? Justify your answer.   

b) For which values of r (if any) does this system of equations have infinitely many solutions? Justify your answer.

8.  Let

a) What is A's characteristic polynomial?               

b) What are A's eigenvalues?    

c) For each eigenvalue identified in (b), find a corresponding eigenvector. In each case, show your working.       

9. What are the determinants of the following matrices, and which ones are invertible? Justify your answers.

10. Let

a) Find the eigenvalues of A, and for each eigenvalue find a corresponding eigenvector.

b) Find an invertible matrix P and a diagonal matrix D such that P^-1AP = D (or if no such P and D exist, explain why not).  

c) Find A⁶.