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Take-Home Assignment 2

MAT223 - Winter 2025

Please make sure to read the Assignment Instructions posted on Quercus to find out how and when to submit your assignment, what is expected of you, etc. Note that there is one optional question, and that not all questions will be marked.

1 Read Example 4.2.14 in the textbook (It starts “Find the shortest distance between the nonparallel lines”), and then complete the problem below.

Let L1 and L2 be the following lines in R3:

1.1 Show that L1 and L2 do not intersect, and are not parallel.

1.2 Find the shortest distance between L1 and L2.

2 Determine if the statement below is True or False.

If it’s True, explain why. If it’s False explain why not, or give an example demonstrating why it’s false with an explanation. A correct choice of “True” or “False" with no explanation will not receive any credit.

True or False: Let L be a line in R3 with direction vector d, and let P be a plane with normal vector n. If n · d ≠ 0, then the line L intersects the plane P (i.e. they have at least one point in common.)

3 3.1 Show that if A is a 4 × 4 diagonalizable matrix with characteristic polynomial CA(x) = x 3 (x − 1), then rank(A) = 1.

3.2 Find an example of a matrix A as described in the previous question. Make sure to show that it has the required properties (i.e. that A is diagonalizable, with CA(x) = x 3 (x − 1).)

4 Read the following definition carefully, and then answer the questions below.

Definition   We define a (square) n × n matrix to be sneaky if there is some real number λ, so that all of it’s diagonal entries are equal to λ, all of its entries directly above the diagonal are equal to 1, and all other entries are equal to 0. That is, it is a matrix of one of these forms, with the pattern continuing for larger n:

Before you continue, you may wish to explore this definition by creating some examples of sneaky matrices.

4.1 Show that the rank of a sneaky matrix is n if λ ≠ 0, and n − 1 if λ = 0.

4.2 Show that if A is an n × n sneaky matrix, then it has exactly one basic eigenvector.

5 Read the following definition carefully, and then answer the questions below.

Def   We define a (square) n × n matrix A to be insufferable if there is some integer k > 0, so that it A k is the zero matrix, but A itself is not the zero matrix.

5.1 True or False: If A is insufferable, then A is non-invertible.

5.2 Determine for which (if any) values of λ a sneaky matrix A (with value λ on it’s diagonal) must be insufferable.

6 Do not hand in: This question explores a simple, but satisfying application of eigenvectors. Suppose that you are the manager of a truck rental company, "You-May-Haul". You rent trucks to people from one of your three locations, which are in Mississauga, Toronto, and Barrie. When someone rents a truck, they have three options for where to leave it - at the location they rented it from, or one of the other locations. After collecting data for a year, you have determined the following patterns among your customers:

• Trucks rented in Mississauga are returned to (M)ississauga 20% of the time, to (T)oronto 40% of the time, and to (B)arrie the remaining 40% of the time.

• Trucks rented in Toronto are returned to these cities (in the same order, Mis-sissauga, Toronto, Barrie) in these proportions: 30%, 40%, 30%, respectively.

• And trucks rented in Barrie are returned in these proportions: 40%, 40%, 20%, respectively.

Notice that if you had M, T and B trucks located at your stores in these cities on a particular day, then (assuming trucks are rented for just one day at a time), the next day, you will have A ·  many trucks in the three locations, where A is a particular 3 × 3 matrix involving the information presented above.

6.1 Determine the matrix A.

6.2 Determine the basic eigenvectors for A.

6.3 Explain why, if you were to pick a basic eigenvector for eigenvalue λ = 1, with only positive integer entries, and with exactly 40 trucks in Toronto, this would represent a good number of trucks to locate at each of your stores (assume two things: first, as the owner, you wish to minimize the amount to which you need to move trucks between your locations; and second, that you need to have at least 40 trucks in your Toronto location to meet regular demand.) Finally, in this scenario, how many trucks would you have located in Mississauga and Barrie?