Pettofrezzo text: 3.4 - 2, 4, 7. 3.5 - 6. 4.1 - 2, 4, 6, 8, 9. 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. Describe each elementary matrix as either a reflection, shear, or dilation/magnification.
2. For each of the three matrices below, find
matrices P and D that diagonalize the matrix. That is, for
each matrix A, find a diagonal matrix D, such that A = PDP-1.
For each matrix A, use your results to calculate A10
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
+ 4bc 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. Every room has four doors,
one on each wall. The rooms are connected through the
walls. There are no connections between rooms through the
corners if the rooms. 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 chances 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 3 by 3 matrix that will rotate a
3-dimensional point (v1, v2, v3) by 30 degrees about the
y-axis. Then move the point 4 units right (x-axis) and 3
units 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 Av + B, where A is 3 by 3, v = (v1, v2, v3) is 3 by
1, and B is 3 by 1.