Reference no: EM13909748
1. Which of the following maps T : R3 → R2 are linear transformations? Justify your answers.
![275_img1.png](https://secure.expertsmind.com/CMSImages/275_img1.png)
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
![2356_img8.png](https://secure.expertsmind.com/CMSImages/2356_img8.png)
Using the method you outlined in (a), either find A's inverse, or else demonstrate that A has no inverse.
3. Let
![262_img9.png](https://secure.expertsmind.com/CMSImages/262_img9.png)
Is a invertible? Justify your answer.
4. Let
![319_img2.png](https://secure.expertsmind.com/CMSImages/319_img2.png)
a) Write down AT, the transpose of A.
b) Calculate |A| and |AT|.
c) Calculate AAT.
d) Calculate |AAT|.
5. Let
![1437_img3.png](https://secure.expertsmind.com/CMSImages/1437_img3.png)
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
![2319_img4.png](https://secure.expertsmind.com/CMSImages/2319_img4.png)
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
![1040_img5.png](https://secure.expertsmind.com/CMSImages/1040_img5.png)
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.
![1976_img6.png](https://secure.expertsmind.com/CMSImages/1976_img6.png)
10. Let
![2027_img7.png](https://secure.expertsmind.com/CMSImages/2027_img7.png)
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.