Instructions You can submit solutions either by e-mail or on paper. Justify each step. Partial solutions will get partial credit. Each problem is 25 points.

Problem 1. Prove the following statements:

a) If the product of two symmetric matrices A and B of the same size is symmetric then AB=BA.

b)Conversely, if A and B are symmetric matrices of the same size and AB=BA then AB is symmetric.

Problem 2. For which values of the parameter a the following system of equations has a) exactly one solution, b) no solutions, c) infinitely many solutions:

     		       ax + (a+1)y + (a+2)z + t= 1
     		           2ax + ay + 2az + 3t = a+1
      			      3x + 4y + 5z + t = a-2
      			        x + ay + z + t = 5

Problem 3. Find the determinant of the following n by n matrix:

Problem 4. Find all values of a such that in the solution of the following system of equations, x*y=z².

 			                ax+y+z = a
 			                x+ay+z = a+1
    				        x+y+az = a+2