Pettofrezzo text: 3.4 - 2, 4, 6. 3.5 - 6. 4.1 - 2, 4, 6, 8, 10. 4.3 - 1. 4.4 - 2, 4, 6, 7. 4.5 - 2, 4, 6, 7

1. What is the 2x2 matrix that rotates shapes by 60 degrees counterclockwise? Show how to write this matrix as a product of elementary matrices. Explain whether each elementary matrix is a reflection, shear, or dilation/magnification, and in what directions.

2. Find matrices P and D that diagonalize each
matrix A below. That is, find a diagonal matrix D, such
that A = PDP^{-1}. Use your result to calculate A^{10}.

2 0 -2
.25
.9
1
0

0 3 0
.75
.1
-1
2

0 0 3

3. Recall
that an *n* by *n* matrix is diagonalizable if it
has *n* distinct eigenvalues. Prove that the general 2 by 2 matrix

*a b*

*c d*

is diagonalizable if (*a*-*d*)^{2}
+ 4*bc* is positive.

4. Prove that the eigenvalues of a
triangular matrix are the same as the entries of the main
diagonal.

5. Two matrices A and B are said to be similar if there is an
invertible matrix C such that AC = CB.

a. Prove that if X is similar to Y and
Y is similar to Z, then X is similar to Z.

b. Prove that two similar matrices have
the same determinant and the same eigenvalues.

6. In the RI/CT/MA area, 5% of MA residents move to CT each year, and 5% move to RI. In CT each year, 15% of the residents move to MA, and 10% to RI. In RI each year, 10% move to MA and 5% to CT. Assuming that no one moves in or out of these 3 states from an outside state, model this as a Markov Process, and calculate what percent of the total population live in each state after a long period of time. Note that it does not matter how the population distribution begins.

7. Imagine that a mouse is in a maze with 9 "rooms" that
looks like a Tic-Tac-Toe board. The rooms are connected
through the edges of the board but not diagonally. For
example, the center room connects right, left, above and
below. The outer rooms have doors that exit the maze.
The side rooms have one exit door each and the corner rooms have
two. Once the mouse exits the maze it is free. Assume that
the chances of moving into any new room from a given room, as well
as the chance of exiting the maze or staying put are all
equal. Thus your matrix has one absorbing state.
Assuming the mouse is dropped into the center room to start off,
and that it makes a random move every minute.

a. Calculate the expected number of moves before the mouse
exits the maze.

b. Calculate the expected number of times that the mouse
will be in each given room before it exits.

c. If all the exit gates are closed, so that there is no
absorbing state, calculate the chances of being in each of the 9
rooms after the probabilities converge.

You should use an online matrix calculator, or Maple, or Matlab
to help you do your calculation.

8. Assume you are trying to decode a matrix cipher that your intelligence people have determined is encoded in blocks of 3 characters using a 3x3 matrix with (0, 0, 0) shift. Decode HPAFQGGDUGDDHPGODYNOR if the first 9 letters are known to decode to "IHAVECOME". Note: Assume that A is 1, B is 2, ... and Z = 0.

9. Find the best fitting line of the following points using the method of linear regression (least squares) we did in class. The points are: (2, 5) (3, 6) (6, 12) (8, 15) (11, 19).

10. Write down a 3x3 matrix that will rotate a picture 30
degrees about the y-axis. Then move it 4 coordinates right
(x-axis) and 3 coordinates down (z-axis). Then stretch it in
all dimensions by a factor of 2. You should write each piece
of the transformation as a 3x3 matrix, and combine them together
with appropriate multiplications and/or additions, to get a single
transformation Ax + B, where A is 3 by 3, x is 3 by 1, and B is 3
by 1..