University of Wisconsin-Madison Mathematics Department

Math 340 Elementary Matrix and Linear Algebra Lecturer: Arun Ram

Fall 2007

Homework 1: Due September 12, 2007

1. Define matrix, equal matrices, matrix addition, scalar multiplication, matrix multiplication, and transpose.
   
   
2. Do p. 19 problems 1, 6,7,8,9.
   
   
3. Define vector and linear combination. Do p. 20 problems10 and 11.
   
   
4. Define dot product, negative, zero, inverse, invertible, symmetric and skew symmetric.
   
   
5. Write the linear systems on p.8-9 problems 5-14 in matrix form.
   
   
6. Prove: If A and B are nxm matrices then A+B=B+A.
   
   
7. Prove: If A, B and C are nxm matrices then A+(B+C)=(A+B)+C.
   
   
8. Prove: If A is an mxn matrix, B is an nxp matrix and C is an pxr matrix then (AB)C=A(BC).
   
   
9. Prove: If A and B are mxn matrices and C is an nxp matrix then (A+B)C=AC+BC.
   
   
10. Prove: If C is an mxn matrix and A and B are nxp matrices then C(A+B)=CA+CB.
    
    
11. Prove: If r and s are numbers and A is a matrix then r(sA) = (rs)A.
    
    
12. Prove: If r and s are numbers and A is a matrix then (r+s)A = rA+sA.
    
    
13. Prove: If r is a number and A and B are matrices then r(A+B) = rA+rB.
    
    
14. Prove: If r is a number and A and B are matrices then A(rB) = r(AB) = (rA)B.
    
    
15. Prove: If A is a matrix then (A^t)^t = A.
    
    
16. Prove: If A and B are mxn matrices then (A+B)^t = A^t + B^t.
    
    
17. Prove: If A is an mxn matrix and B is an nxp matrix then (AB)^t = B^tA^t.
    
    
18. Prove: If r is a number and A is a matrix then (rA)^t = rA^t.
    
    
19. Give an example of two 5x5 matrices A and B such that AB is not equal to BA.
    
    
20. Give an example of two 5x5 matrices A and B such that no entries of A or B are 0, A is not equal to B, and AB = BA.