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 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
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. The rooms are connected through the edges of the board but not diagonally. For example, the center room connects right, left, above and below. Assume that the chances of moving into any new room from a given room, as well as the chance of staying put are all equal. Calculate the chances that the mouse will be in a given room at a given time. Feel free to use a tool like Maple or MatLab, to help the solve the resulting 9x9 array.
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 least squares. 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
degress
about the y-axis, move it 4 coordinates right (x-axis) and 3
coordinates down (z-axis),
and
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 the
overall
3x3 matrix transformation.