Everyone must take both the Proctored Test and the Take-home Test.¶
A proctored test Friday, June 26, 2026 at 1:00-2:50 pm, in room 261 of Jabara Hall.
A take-home test posted on Tuesday, June 23, 2026 that must be submitted to Blackboard in a single pdf by 11:59 pm, CDT, Saturday, June 27, 2026.
Exam¶
- The exam consists of both tests. Your exam score is the average of the percentage scores of each test.
- You need a 50% or better on each test to pass this exam. The 50% requirement for each individual test is an additional requirement to the conventional 10-point grading scale. This requirement exists to ensure that everyone demonstrates mastery of the coursework without referring to notes or help.
Proctored Test¶
- You will have 110 minutes to complete the proctored test.
- There are no notes, electronic devices, note cards, books, or any aids allowed during the proctored test.
- You may bring a non-graphing, basic scientific calculator for arithmetic. Programmable calculators, tablets, smart watches, smart glasses, devices that can solve a system of equations, devices that can perform differentiation or integration, devices connected to the internet are not allowed. Put them away before the test. Having such a device unstowed during the proctored test results in a zero for the test.
Take-home Test¶
- The take-home test must be turned in on time in a single readable PDF to receive credit.
- You can use technology to check your work. You must still show all of your work. I am grading the work you complete to compute the answer, not the numerical result. You must show all of the steps you performed using the techniques we learned in these chapters. "I computed it with my software" is a zero.
- There is no time limit other than the due date and time.
- You must do your own work.
I am grading mastery of the techniques taught in this course. Use the techniques from chapters 1-3 for this test.
| Criteria | Excellent | Good | Satisfactory | Needs Improvement |
|---|---|---|---|---|
| Understanding (25%) | Demonstrates a comprehensive understanding of concepts, uses precise terminology, and explains ideas clearly. | Shows a good understanding of concepts, minor inaccuracies in terminology, explanations are generally clear. | Basic understanding of concepts, some errors in terminology, explanations need more clarity. | Limited understanding, frequent errors in terminology, explanations are unclear. |
| Computational Accuracy (25%) | All calculations are correct, logical, and well-organized. | Minor calculation errors, overall approach is sound. | Significant calculation errors affecting results. | Calculations are mostly incorrect or unclear. |
| Clear Mathematical Presentation (25%) | Presents work in a clear, organized, and logically structured manner, using proper mathematical notation and formatting. | Presentation is generally clear, with minor errors in notation or vocabulary. | Presentation is somewhat unclear or lacks organization, with more frequent errors in notation or vocabulary. | Presentation is difficult to follow, with significant errors in notation or vocabulary. |
| Linear Algebra (25%) | Explicitly connects linear algebra concepts to the solution, highlighting their relevance and demonstrating understanding of their role. | Makes some connections to linear algebra concepts, but could be more explicit or detailed. | Connections to linear algebra concepts are unclear or superficial. | Fails to consistently or accurately connect linear algebra concepts to the solution. |
To get any credit for solving a linear system one must¶
- possibly set up the linear system
- write out the augmented matrix
- reduce the matrix to one of the following forms:
- upper triangular form
- row echelon form
- reduced row echelon form using row operations.
- use backward substitution
I realize there are other methods of solving these problems because they are small. They are only small to allow one to complete the exercise with a reasonable amount of effort.
To get any credit for your answers¶
You must use proper notation:
- display a solution as a column vector
- angle brackets are used to indicate that the list of numbers is really a column
- square or round brackets enclosing a vertical (column) list of numbers and/or variables
- square or round brackets enclosing a horizontal list of numbers with $^T$ (transpose).
- $(2,1)$ is a point in a $2$-dimensional space, not a vector in a vector space.
- $x_1=2$, $x_2=1$ is not a vector.
- The correct answer is
$$ \begin{bmatrix} 2 \\ 1 \end{bmatrix} = \left(\begin{matrix} 2 \\ 1 \end{matrix}\right) = \begin{bmatrix} 2 & 1 \end{bmatrix}^T = \left(\begin{matrix} 2 & 1 \end{matrix}\right)^T = \langle 2, 1 \rangle $$
I understand that there are differences among textbooks and professors about notation. When providing solutions to a homework problem or an exam, it is the student's job to clearly communicate their mastery of the concepts and ideas. Inventing one's own notation does not communicate mastery. Employing conflicting notation also does not commmunicate mastery; only ambiguity. I will not award points for ambiguous or non-compliant notation.
To get any credit for computing a determinant¶
A $2\times 2$ matrix may be evaluated using a formula, $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$.
To compute the determinant of a larger $n\times n$ matrix ($n>2$), one must use the ten properties of determinants used in the course
There are a limited number of places that the Laplace Expansion is allowed:
- Computing the determinant of a $2\times 2$ matrix.
- When a row or column has only one nonzero entry.
- When computing the characteristic polynomial if necessary.
Chapters Covered in First Exam¶
The first proctored and take-home test covers chapters 1, 2 and 3 from the online notes and the textbook (eBook). You should study:
General Review¶
Any question from the WebAssign homework, Problem Sets, Exercises, or Review Exercises (at the end of chapters 1, 2, and 3).
Any question similar to a question from the WebAssign homework, Problem Sets, Exercises, or Review Exercises (at the end of chapters 1, 2, and 3).
Review Questions¶
The online notes have many examples and exercises in each section for study.
The textbook (eBook) includes review exercises at the end of each chapter. In particular, the textbook includes true/false questions. This is a form of vocabulary exercise that appears in all proctored tests.
The textbook (eBook) contains a 1-3 Cumulative Test at the end of chapter 3 for you to study.
The problem sets introduce important applications of concepts in each chapter.
Well-prepared means:¶
Review Topics¶
- You have your own notes,
- You have memorized the basic formulas, definitions and properties
- You practiced the methods of computing
- Gaussian Elimination
- Elementary Row Operations and Elementary Matrices
- The LU decomposition
- The CR decomposition
- The solution to a linear system
- Determinants
- Adjugate
- Cramer's Rule
- The inverse of a nonsingular matrix
- The inverse of a $2\times 2$ nonsingular matrix using the adjugate
Answering True/False Questions¶
In a true/false question, a mathematical statement is provided. You must determine whether the statement is true or false. True here means always true. In mathematics, a statement that is sometimes true and sometimes false is called false. Your answer has two parts:
- The verdict (1 point). Write TRUE or FALSE.
- The justification (2 points).
- If TRUE, prove it in general. Give a derivation, proof, or cite a definition or theorem from the notes. Here a general argument with variables is exactly what you should provide. An example only shows the statement is true once. You need to show it is always true.
- If FALSE, give a counterexample. One specific instance with actual numbers (concrete vectors/matrices with numerical entries) in which the hypotheses hold but the conclusion fails. Show the computation.
(Self-Check: Could someone evaluate my example with no choices left to make? If a letter or variable remains that could be anything, it is not yet a counterexample; pick a number.)
Example 1 - TRUE¶
- For $m\times n$ matrices $A$ and $B$, $A + B = B + A$.
Answer¶
TRUE
$A + B = [a_{ij}] + [b_{ij}] = [ a_{ij} + b_{ij} ] = [b_{ij} + a_{ij}] = [b_{ij}] + [a_{ij}] = B + A$
Example 2 - FALSE¶
- For $n\times n$ matrices $A$ and $B$, $AB = BA$.
Answer¶
FALSE
Let $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$, and $B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$. Then
$$ \begin{align*} AB &= \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \\ \\ BA &= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \neq AB \end{align*} $$
Exercise 1 - TRUE/FALSE¶
- If a system of linear equations has more unknowns than equations, then it has infinitely many solutions.
View Solution
FALSE. With more unknowns than equations a unique solution is impossible, but the system can still be inconsistent. Counterexample (2 equations, 3 unknowns): $$ x_1 + x_2 + x_3 = 0, \qquad x_1 + x_2 + x_3 = 1. $$ Subtracting gives $0 = 1$, so the system has no solutions, not infinitely many.
- A homogeneous linear system $A\mathbf{x}=\mathbf{0}$ is always consistent.
View Solution
TRUE. The zero vector is always a solution, since $A\mathbf{0}=\mathbf{0}$. A system with at least one solution is consistent.
- If $A$ is a square matrix and $A\mathbf{x}=\mathbf{0}$ has only the trivial solution, then $A$ is nonsingular.
View Solution
TRUE. For a square matrix this is one of the equivalent characterizations of nonsingularity: $A\mathbf{x}=\mathbf{0}$ has only the trivial solution $\iff$ $\operatorname{rref}(A)=I$ $\iff$ $A$ is nonsingular.
- If $A$ and $B$ are symmetric $n\times n$ matrices, then $AB$ is symmetric.
View Solution
FALSE. $(AB)^T = B^T A^T = BA$, which equals $AB$ only when $A$ and $B$ commute. Counterexample with symmetric matrices: let $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$. Then $$ AB = \begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}, $$ whose $(1,2)$ and $(2,1)$ entries are $-2$ and $2$. Since $AB \neq (AB)^T$, it is not symmetric.
- For all $n\times n$ matrices $A$ and $B$, $(A+B)^2 = A^2 + 2AB + B^2$.
View Solution
FALSE. Expanding, $(A+B)^2 = A^2 + AB + BA + B^2$, which equals $A^2+2AB+B^2$ only when $AB=BA$. Counterexample: let $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$. Then $$ (A+B)^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad A^2 + 2AB + B^2 = 2AB = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}, $$ and these are not equal.
- For all $n\times n$ matrices $A$ and $B$, $(AB)^T = B^T A^T$.
View Solution
TRUE. The transpose of a product reverses the order of the factors; this is a standard property of the transpose, valid for all conformable $A$ and $B$.
- If $A$ and $B$ are invertible $n\times n$ matrices, then $AB$ is invertible and $(AB)^{-1} = B^{-1}A^{-1}$.
View Solution
TRUE. $$ (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I, $$ and similarly $(B^{-1}A^{-1})(AB) = I$. So $AB$ is invertible with inverse $B^{-1}A^{-1}$.
- If $A$ and $B$ are invertible $n\times n$ matrices, then $A+B$ is invertible.
View Solution
FALSE. Counterexample: let $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$. Both are invertible, but $$ A + B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, $$ which is singular.
- For all $n\times n$ matrices $A$ and $B$, $\det(AB) = \det(A)\det(B)$.
View Solution
TRUE. The determinant is multiplicative: $\det(AB)=\det(A)\det(B)$ for all square matrices of the same size.
- For an $n\times n$ matrix $A$ and a scalar $c$, $\det(cA) = c\,\det(A)$.
View Solution
FALSE. Scaling all $n$ rows by $c$ scales the determinant by $c^n$, so $\det(cA)=c^n\det(A)$. Counterexample: $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $c = 2$. Then $$ \det(2A) = \det\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = 4, \qquad c\,\det(A) = 2(1) = 2, $$ and $4 \neq 2$.
- An $n\times n$ matrix $A$ is nonsingular if and only if $\det(A) \neq 0$.
View Solution
TRUE. This is one of the equivalent conditions for nonsingularity: $A$ is invertible $\iff \det(A)\neq 0$.
- Swapping two rows of a square matrix does not change its determinant.
View Solution
FALSE. A single row swap multiplies the determinant by $-1$. Counterexample: $\det\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = 1$, but swapping the rows gives $\det\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = -1$.
- The product of two elementary matrices is always an elementary matrix.
View Solution
FALSE. An elementary matrix differs from $I$ by a single elementary row operation, but a product generally encodes two. Counterexample: let $E_1 = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$ (add $2\times$ row 1 to row 2) and $E_2 = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$ (add $3\times$ row 2 to row 1). Then $$ E_1 E_2 = \begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}, $$ which cannot be obtained from $I$ by one row operation, so it is not elementary. (It is still invertible — products of elementary matrices always are — just not elementary.)
- For every $n\times n$ matrix $A$, $\;A\,\operatorname{adj}(A) = \det(A)\,I$.
View Solution
TRUE. This is the fundamental adjugate identity. When $\det(A)\neq 0$ it rearranges to $$ A^{-1} = \frac{1}{\det(A)}\operatorname{adj}(A), $$ which is the formula behind both the adjugate method for the inverse and Cramer's rule.
Multiple choice questions on the proctored test are a quicker and easier kind of True/False question. Like True/False questions, these are mainly vocabulary questions. In each multiple choice question,
- check the box of a statement that is true, and
- leave the box of a false statement unchecked.
You do not need to show any justification. Simply check the true statements.
Example 3 - Multiple Choice¶
- If $A$ is an $n\times n$ matrix, then check the following that must be true.
$\square\ $ The linear system $A\mathbf{x}=\mathbf{0}$ is consistent.
$\square\ $ The reduced row echelon form of matrix $A$ is $I$ if and only if matrix $A$ is nonsingular.
$\square\ $ The linear system $A\mathbf{x}=\mathbf{b}$ is consistent only if it has a unique solution.
$\square\ $ If $AB=O$, then $A=O$ or $B=O$.
$\square\ $ If $A$ is symmetric, then $A^T$ is symmetric.
If you check the first, second, and last boxes; and leave the third and fourth boxes unchecked, then you get 5/5.
Exercise 2 - Multiple Choice¶
- If $A$ is an $n\times n$ matrix, then check each of the following that must be true.
$\square\ $ $\det(A^T) = \det(A)$.
$\square\ $ $\det(A+B) = \det(A) + \det(B)$ for every $n\times n$ matrix $B$.
$\square\ $ If $A$ is nonsingular, then $\det(A^{-1}) = \dfrac{1}{\det(A)}$.
$\square\ $ If one row of $A$ is a scalar multiple of another row, then $\det(A)=0$.
$\square\ $ If $\det(A)=0$, then $A\mathbf{x}=\mathbf{0}$ has only the trivial solution.
View Solution
If you check the first, third, and fourth boxes; and leave the second and fifth boxes unchecked, then you get 5/5.- If $A$ is an $n\times n$ matrix, then check each of the following that must be true.
$\square\ $ If $A$ is nonsingular, then $A\mathbf{x}=\mathbf{b}$ has a unique solution for every $\mathbf{b}$.
$\square\ $ If $A$ is singular, then $A\mathbf{x}=\mathbf{b}$ has no solution.
$\square\ $ If $A$ is nonsingular, then $\left(A^{-1}\right)^{-1} = A$.
$\square\ $ If $A^2 = I$, then $A = I$ or $A = -I$.
$\square\ $ The inverse of a nonsingular matrix is unique.
View Solution
If you check the first, third, and last boxes; and leave the second and fourth boxes unchecked, then you get 5/5.We will be practicing the skills used in the first three chapters throughout the rest of the course. It is alright to struggle with the first exam. I expect your test scores to get much better and I will only average your Exam 1 score into your grade if it raises your grade at the end of the semester.
- I am setting expectations
- You are finding out what the tests look like
The first exam is for my information more than yours. You will get better with practice and there will come a time when this all seems as easy as the algebra of scalars.
Unlike your previous Mathematics courses, there are a lot of definitions and concepts to memorize. Give yourself some time. Be patient and keep up the good work.
