University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 3: Due September 26, 2007

1. Define row echelon form and explain how to row reduce (any) matrix to row echelon form.
   
   
2. Define permutation, inversion set, length of a permutation, even permutations and odd permutations and give examples.
   
   
3. Define the determinant of a matrix and give an example.
   
   
4. Define the trace of a matrix and give an example.
   
   
5. Show that det(AB) = det(A)det(B).
   
   
6. Show that tr(A+B) = tr(A)+tr(B).
   
   
7. Find the traces and determinants of the elementary matrices.
   
   
8. Suppose that det(A)=5. Show that det(A^-1)=1/5.
   
   
9. Show that the determinant of an upper triangular matrix is the product of its diagonal entries.
   
   
10. Define n! and show that the number of permutations in S_n is n!.
    
    
11. Let A be an nxn matrix. Show that if A is invertible then det(A) is not 0.
    
    
12. Let A be an nxn matrix. Show that if det(A) is not 0 then A is invertible.
    
    
13. Show that tr(AB)=tr(BA).
    
    
14. Define minor and cofactor.
    
    
15. Compute the determinants of the matrices in problems 10, 11, 12 on p. 125 by the definition of the determinant.
    
    
16. Compute the determinants of the matrices in problems 10, 11, 12 on p.125 by expansion down the first column.
    
    
17. Compute the determinants of the matrices in problems 10, 11, 12 on p. 125 by first doing row reduction.
    
    
18. Calculate the cofactor matrix for each of the matrices in problems 10, 11, 12 on p. 125.