MAT 223 Mike University Algebra System of Linear Equations Worksheet

Assignment 1MAT 2 2 3 – Wi n t e r 2022
1
Let a and b be arbitrary real numbers.
Consider a system of linear equations whose augmented matrix is as follows:


a 0 0 a2 1 b


 0 0 b 1 b2 a  .
0 0 0 0 0 0
Determine the solution(s) (if any) to the corresponding system of equations. Write
solution(s) (if there are any) in parametric form. If there are no solutions, explain
why not.
• Keep in mind that a and b are arbitrary (they could be positive, negative or
zero potentially.)
• Hint: you should consider different cases depending on the values of a and
b (for example, if a = 0 or a ̸= 0). Carefully describe the distinct cases and
what the solution(s) is/are (if any) in each case. Clearly organize your work.
2
Determine if the statement below is True or False.
If it’s True, explain why. If it’s False explain why not, or simply give an example
demonstrating why it’s false (with a brief explanation). A correct choice of “True” or
“False” with no explanation will not receive any credit.
True or False: For any 3 × 5 system of linear equations, either it has infinitely
many solutions or no solutions.
1
3
Read Section 1.4 (page 27) in our Course Textbook, https://lila1.lyryx.com/
textbooks/OPEN_LAWA_1/marketing/Nicholson-OpenLAWA-2021A.pdf
Then consider the following network, and answer the questions that follow:
3.1
Write down a system of linear equations describing the flow through this network,
using the variables f 1 , …, f 6 . (Don’t solve the system or do any other work for this step.)
3.2
The RREF of the augmented matrix corresponding to the system you produced in
the first step is (or at least should) be:







1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0 1
35
0 −1 −5
0 0
35
55
0 1
1 −1 −30







Write out the general solution to the system of equations from the first step in this
question using this information.
4
3.3
Can we determine an exact value for the flow into junction B? If so, what is it? If
not, explain why not.
3.4
Suppose that the into junction C is restricted to being between 26 and 28 (inclusive).
What range of possible values can there be for the flow out of junction E? Assume
that all variables cannot be negative.
Read the following definitions carefully, and then answer the questions below.
Note that one of the definitions appears on Tutorial Worksheet 1, so you should make sure
to go to tutorial this week to make sure you are prepared for this assignment!
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Definition
We define a matrix to be ephemeral if the number of non-zero entries 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 non-zero entries is at most 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 variables, 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 at most min(m, n).
Before you continue, you may wish to explore this definition by creating some
examples of ephemeral or 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.
4.1
Can a 2 × 2 matrix be both ephemeral and solid?
4.2
Can a 2 × 2 matrix be neither ephemeral nor solid?
4.3
Can a matrix with at least three rows and at least three columns be both ephemeral
and solid?
4.4
Can a matrix with at least three rows and at least three columns be neither ephemeral
nor solid?
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