Exam 2 Fall 2009

Name: ______This is a closed book exam. You may use a calculator and the formulas handed out with this exam. You may find that your calculator can do some of the problems. If this is so, you still need to show how to do the problem by hand, even if you use a calculator to check your work. On all the problems, show all work and explain any reasoning which is not clear from the computations. (This is particularly important if I am to be able to give part credit.) Turn in this exam with your answers. However, don't write your answers on the exam itself; leave them on the pages with your work. Also turn in the formulas; put them on the formula pile.

1.(20 points) Use the method discussed in class for finding the inverse of a matrix to find the inverse of . If you do your work correctly, all your numbers should be integers, i.e. no fractions. You need to show your work. Just having your calculator find the inverse is not acceptable.

2.Suppose the formulas

2x - 3y = u

5x + ay = v

relate the two variables u and v to the two variables x and y. Here a is some fixed constant.

a.(10 points) Find formulas for x and y in terms of u, v, and a. Your answer should be in the form

x = some formula in terms of u, v, and a

y = some formula in terms of u, v, and a

b.(10 points) Suppose an increase of u by 1 while leaving v and a fixed causes x to increase by 2. What is a? Explain how you arrive at this answer.

3.(15 points) Let A = . Suppose A is both symmetric matrix and invertible. Is it always true that this implies A-1 is also symmetric? If so, prove it. If not, construct a counterexample.

4.Let A = . Suppose the determinant of A is -5.

a.(10 points) What is the determinant of (6A)-1? Explain why.

b.(10 points) What is the determinant of ATA? Explain why.

5.(20 points) Suppose A = . Use determinants to find the entry of the 2nd row and 3rd column of A-1. Show how to evaluate the determinants by hand even if you have a calculator that will do it.

Solutions to Exam 2 Fall 2009

1.

A-1 =

2.a.The equations can be written as Az = w where A = , z = and w = . Multiply by A-1 to get z=A1w= . So x = , y = .

b.Increasing u by 1 while holding a and v fixed causes x to increase by . So = 2. So a = 4a + 30. So 3a = -30. Thus a = -10.

3.Method 1. Since A is symmetric on has AT = A. So (AT)-1 = A-1. In class we showed that (AT)-1 = (A-1)T. So (A1)T=A-1. Thus A-1 is symmetric.

Method 2. Using the formula for A-1, one has A-1 . Since A is symmetric one has b = c. So A-1 . Since (A-1)12 = (A-1)21, it follows that A-1 is symmetric.

4.a.det( (6A)-1 ) = = = = .

b.det( ATA ) = det( AT ) det( A ) = det( A ) det( A ) = (-5)(-5) = 25.

5.A-1 = where Bij = cofactor of Aij = (-1)i+jdet( minor(Aij) ). So (A-1)23 = = . One has = -4. Expanding by cofactors along the second row gives = (3) (-2) = -6. So (A-1)23 = = .