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Take-Home Assignment 1

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 In this question, a and b are arbitrary real numbers.

For which value(s), if any, of a and b is the system with the following augmented matrix consistent? Determine all solutions for any value(s) of a and b for which the system is consistent. Hint: In order to properly row reduce this matrix, you will need to split your work across several cases (e.g. if a = 0 or a ≠ 0, etc.)

2 Determine if the statements below are 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.

Note: The Week 1 Asynchronous Activities include a video about how to approach True/False questions - check it out if you haven’t yet!

True or False: If an m × n system of equations has infinitely many solutions, then the RREF of the augmented matrix for the system must have a row of 0’s.

True or False: If A and B are 2 × 2 matrices, neither of which is skew-symmetric, then AB is not skew-symmetric.

3 Read the following definitions carefully, and then answer the questions below.

Definition We define a matrix to be virtual if the number of non-zero entries is strictly less than the number of rows or columns, whichever is smaller. i.e. if m is the number of rows, and n is the number of columns, then the number of non-zero entries is strictly less than min(m, n).

Similarly, a matrix is called solid if the number of entries that are equal to 0 is less than or equal to the number of rows or columns, whichever is smaller. i.e. if m is the number of rows, and n is the number of columns, then the number of entries equal to zero is less than or equal to min(m, n).

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

For each of the following questions, if the answer is "yes", create such a matrix and explain why it has (or doesn’t have) the desired properties; if the answer is "no", give an explanation for why.

3.1 Can a 2 × 2 matrix be both virtual and solid?

3.2 Can a 2 × 2 matrix be neither virtual nor solid?

3.3 Can a matrix with at least three rows and at least three columns be both virtual and solid?

3.4 Can a matrix with at least three rows and at least three columns be neither virtual nor solid?

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

Def We define an n × n matrix A to be terminal if A 2 = A.

Note: before diving in to solving the questions below, you may wish to try to create some examples of terminal matrices (say 2 × 2 to keep it simple).

4.1 Suppose that A is terminal. Show that I − A is also terminal. Show that A − I is not terminal unless A = I.

4.2 True or False: if A is m × n and B is n × m and AB = Im, then BA is terminal.

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

Def We define an n × n matrix B to be flat if B 2 = I.

Note: as before, start by trying to create some examples of small flat matrices.

True or False: If B is an n × n flat matrix, then the homogeneous system Bx = 0 has only the trivial solution.

6 Do not hand in: This question investigates Theorem 2 from the Sections 2.1-2.3 pre-class reading (see p.8) by examining a specific example. It is optional; i.e. it will not be marked. But you should do it, as it is an example of how you can understand what a Theorem is saying when you are doing the readings.