Comprehensive Study Notes on Matrix Applications and Matrix Multiplication

Course Logistics and Administrative Guidelines

• Add/Drop Period and Roster Updates:

  • The course roster remains open until Monday to accommodate late additions and drops.
  • Roster updates are not checked hourly during this window.

• Homework Schedules (Cengage):

  • Initial homework assignments are due on September 4.
  • Assignments for the following week are scheduled to be due around the same time.
  • Deadlines are extended for the first week specifically due to the add/drop period and to ensure all enrolled students are registered in the system.
  • Students should begin working on assignments early despite extended deadlines.

• Syllabus Scavenger Hunt:

  • A syllabus scavenger hunt assignment is due on Monday.

• iClicker, App, and Location Troubleshooting Procedures:

  • Technical/Location Errors:
    • If the app displays a location error or prevents access, log out of the application and log back in.
    • Disconnecting from campus Wi-Fi and reconnecting can resolve network-related location errors caused by heavy traffic during the first few weeks of class.
  • In-Class iClicker Failure Protocol:
    • If technical issues persist during an active clicker question, take a photograph of the classroom projection screen (do not include the instructor in the photo).
    • Email the photo to the instructor immediately following class to receive attendance credit for the day.
  • Location Services Rationale:
    • Location tracking is enforced to verify physical classroom presence and prevent absent students from participating remotely via classmate prompts.

• Assessment and Work Expectations:

  • Showing Work:
    • All intermediate arithmetic steps must be written out in full on quizzes and exams.
    • Showing explicit setup steps allows partial credit to be awarded if minor arithmetic errors occur (e.g., miscalculating $0 \times -1$ as $-1$).
    • Answers presented without supporting work are ineligible for credit.
  • Organization:
    • Work must be clearly organized and labeled to facilitate efficient grading.
  • Matrix Labeling:
    • All matrices and final answer matrices must have their rows and columns explicitly labeled every single time.

Section 1.1: Matrix Modeling and Applied Scalar Operations

• Applied Mathematics Context:

  • Course focus is centered on real-world applications and word problems rather than abstract pure mathematics.
  • Word problems require careful reading to extract data accurately into matrix structures.

• Constructing a Matrix from Real-World Data:

  • Scenario Context: Local university stadium vendors track inventory sales for hot dogs, pretzels, and pickles on a stick across home and visitor sides during a conference football game.
  • Raw Data Provided:
    • Home side estimated sales: $250$ hot dogs, $475$ pretzels, $300$ pickles on a stick.
    • Visitor side estimated sales: $200$ pretzels, $550$ pickles on a stick, $325$ hot dogs.
  • Defining Matrix Dimensions and Labels:
    • Matrix Name: Named $F$ (to represent Food).
    • Dimensions: A $2 \times 3$ matrix (2 rows and 3 columns).
    • Rows ($2$): Represent the stadium sides.
      • Row 1 = Home Side ($HS$)
      • Row 2 = Visitor Side ($V$)
    • Columns ($3$): Represent food items.
      • Column 1 = Hot Dogs ($HD$)
      • Column 2 = Pretzels ($P$)
      • Column 3 = Pickles on a Stick ($PS$)
    • Labeling Rule: Distinct abbreviations must be used when category names share initial letters (e.g., $P$ for Pretzels vs. $PS$ for Pickles on a Stick; $HD$ for Hot Dogs vs. $H$ for Home side).
  • Formulating Matrix $F$:
    • Aligning items to designated columns ($HD$, $P$, $PS$):
      • Row 1 (Home): Entry $(1,1) = 250$, Entry $(1,2) = 475$, Entry $(1,3) = 300$.
      • Row 2 (Visitor): Entry $(2,1) = 325$, Entry $(2,2) = 200$, Entry $(2,3) = 550$.
    • Matrix Equation with Labels:             $\begin{matrix} & \begin{matrix} HD & P & PS \end{matrix} \\ \begin{matrix} HS \\ V \end{matrix} & \begin{pmatrix} 250 & 475 & 300 \\ 325 & 200 & 550 \end{pmatrix} \end{matrix}$

• Scalar Multiplication Application (Inventory Increase):

  • Scenario: Vendors prepare for the next home game against a rival by ordering $15\%$ more than sold at the previous game.
  • Mathematical Derivation:
    • An increase of $15\%$ implies total inventory equals original amount plus $15\%$ of original amount.
    • Convert percentage to decimal: $15\% = 0.15$
    • Algebraic formulation: $F + 0.15F = (1 + 0.15)F = 1.15F$
  • Scalar Computation:         $1.15 \times \begin{pmatrix} 250 & 475 & 300 \\ 325 & 200 & 550 \end{pmatrix}$
    • Home Side Hot Dogs: $1.15 \times 250 = 287.5$
    • Home Side Pretzels: $1.15 \times 475 = 546.25$
    • Home Side Pickles on a Stick: $1.15 \times 300 = 345$
    • Visitor Side Hot Dogs: $1.15 \times 325 = 373.75$
    • Visitor Side Pretzels: $1.15 \times 200 = 230$
    • Visitor Side Pickles on a Stick: $1.15 \times 550 = 632.5$
  • Final Matrix Result:         $\begin{matrix} & \begin{matrix} HD & P & PS \end{matrix} \\ \begin{matrix} HS \\ V \end{matrix} & \begin{pmatrix} 287.5 & 546.25 & 345 \\ 373.75 & 230 & 632.5 \end{pmatrix} \end{matrix}$
  • Mathematical Result vs. Practical Reality:
    • Mathematical Answer: Decimals ($287.5$, $546.25$, etc.) represent exact theoretical minimum requirements and constitute the exact required answer on academic assessments.
    • Practical Application: Fractional food items cannot be sold or purchased. Bulk purchasing requirements in practical applications would require rounding up to whole numbers or standard package batch sizes (e.g., ordering $288$ or $300$ items).

Section 1.2: Matrix Multiplication

• Definition and Compatibility Rule:

  • Matrix multiplication involves multiplying two matrices together rather than scaling a matrix by a single constant.
  • Inner Dimension Rule: To multiply matrix $A$ by matrix $B$ (product $AB$), the number of columns in $A$ must equal the number of rows in $B$
  • If matrix $A$ has size $m \times n$ and matrix $B$ has size $p \times q$:
    • The product $AB$ exists if and only if $n = p$
    • If $n \neq p$, the product $AB$ is undefined / not possible.
  • Outer Dimensions Rule: If $n = p$, the resulting product matrix $AB$ has dimensions $m \times q$

• General Multiplication Process:

  1. Determine and write out the dimensions of each matrix.
  2. Compare inner dimensions to verify compatibility.
  3. Determine the size $m \times q$ of the resulting matrix.
  4. Compute the entry in Row $i$, Column $j$ of the product matrix by multiplying each element of Row $i$ of the first matrix by its corresponding element in Column $j$ of the second matrix, then taking the sum of these products.

• Comprehensive Multiplication Example:

  • Given Matrix $A$ ($2 \times 2$):         $A = \begin{pmatrix} 3 & -5 \\ 0 & a \end{pmatrix}$

  • Given Matrix $B$ ($2 \times 3$):         $B = \begin{pmatrix} -1 & b & 11 \\ 2 & -6 & 4 \end{pmatrix}$

  • Part 1: Evaluating Product $AB$:

    • Dimension Check: $A$ ($2 \times 2$) and $B$ ($2 \times 3$).
    • Inner dimensions: $2$ and $2$ (equal; product exists).
    • Resulting Matrix Size: $2 \times 3$ ($2$ rows, $3$ columns, $6$ total entries).
    • Row 1 Computations:
      • Row 1, Column 1: $(3 \times -1) + (-5 \times 2) = -3 + (-10) = -13$
      • Row 1, Column 2: $(3 \times b) + (-5 \times -6) = 3b + 30$
      • Row 1, Column 3: $(3 \times 11) + (-5 \times 4) = 33 + (-20) = 13$
    • Row 2 Computations:
      • Row 2, Column 1: $(0 \times -1) + (a \times 2) = 0 + 2a = 2a$
      • Row 2, Column 2: $(0 \times b) + (a \times -6) = 0 + (-6a) = -6a$
      • Row 2, Column 3: $(0 \times 11) + (a \times 4) = 0 + 4a = 4a$
    • Intermediate Step Setup:             $AB = \begin{pmatrix} (3)(-1) + (-5)(2) & (3)(b) + (-5)(-6) & (3)(11) + (-5)(4) \\ (0)(-1) + (a)(2) & (0)(b) + (a)(-6) & (0)(11) + (a)(4) \end{pmatrix}$
    • Final Simplified Product Matrix:             $AB = \begin{pmatrix} -13 & 3b + 30 & 13 \\ 2a & -6a & 4a \end{pmatrix}$

  • Part 2: Evaluating Product $BA$:

    • Dimension Check: $B$ ($2 \times 3$) and $A$ ($2 \times 2$).
    • Inner dimensions: $3$ (columns of $B$) and $2$ (rows of $A$).
    • Since $3 \neq 2$, inner dimensions do not match.
    • Conclusion: Product $BA$ is not possible (undefined).

• Properties of Matrix Operations:

  • Non-Commutativity: Matrix multiplication is NOT commutative ($AB \neq BA$ in general).
    • Order cannot be altered.
    • Changing operational order can alter dimension compatibility or produce different results entirely.
  • Associativity: Matrix multiplication is associative: $(AB)C = A(BC)$, provided product compatibility is maintained and matrix sequence remains strictly ordered.
  • Distributivity: Matrix multiplication distributes over addition while preserving position:
    • Left Distributive Property: $C(A + B) = CA + CB$
    • Right Distributive Property: $(A + B)C = AC + BC$
    • Caution: Relative positions must be maintained ($C(A+B) \neq AC + BC$). Parentheses cannot be added or removed arbitrarily.