University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 2: Due September 19, 2007

1. Do p. 30-31 problems 1-16.
   
   
2. Show that there is a unique mxn matrix A such that A+B = B for all mxn matrices B.
   
   
3. Show that there is a unique nxn matrix A such that AB for all nxn matrices B.
   
   
4. Let A be a matrix. Show that there is a unique matrix B such that B+A=0.
   
   
5. Give an example of a nonzero 5x5 matrix A such that there does not exist a 5x5 matrix B with BA=1.
   
   
6. Prove: If A is an nxn matrix and A^-1 exists then A^-1 is unique.
   
   
7. Prove: If A and B are nxn matrices and A^-1 and B^-1 exist then (AB)^-1 exists.
   
   
8. Let A be a nxn matrix and assume that A^-1 exists. Show that (A^t)^-1 = (A^-1)^t.
   
   
9. Explain how to use matrices to solve a system of linear equations. Give some examples.
   
   
10. Define elementary matrices, elementary row operations, and elementary column operations.
    
    
11. Find the inverses of the elementary matrices. Give an example for each case.
    
    
12. Let A be an nxn matrix. Explain how to write A as a product of elementary matrices.
    
    
13. Let A be an nxn matrix. Explain how to find A^-1.
    
    
14. Write the matrix of p. 129 problem 2(d) as a product of elementary matrices.
    
    
15. Find the inverse of the matrix of p. 129 problem 2(d).
    
    
16. Do p. 125 problems 10, 11, 12.
    
    
17. Solve the linear systems in p.114 problem 6.