Test 1.

You have 1 hour and 20 min to solve these problems. Show your work. You should give not just an answer but also a complete justification of it. Each problem is 20 points worth.

1. Prove that if A and B are invertible matrices of the same size then a) AB is invertible and b) the inverse of AB is equal to the inverse of B times the inverse of A.

2. Is the product of two symmetric matrices of the same size

  • a) always symmetric?
  • b) never symmetric?

    3. For which values of the parameter a is the following matrix invertible? 

                              a 1 1
                          a a 1
                          a a a
    In the case when it is invertible, find the inverse.

    4. Represent the following matrix as a product of elementary matrices: 

                              3 4
                          5 6


    5. Find the general solution of the following system of equations: 

                  2x+2y+3z+2t = 3
              x+y+2z+t+u  = 2
              x+y+t-u     = 0

    Good luck!