Matrix Foundations Guided Notes
Matrix Foundations
Guided Lecture Notes • Unit 3: Matrices
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1 What is a Matrix?
Definition & Structure
A matrix (plural: matrices) is a rectangular arrangement of numbers organized in and used to collect and display data.
Each individual value in a matrix is called an (or component).
Real-World Data Table
| Store | Chicken | Fries |
|---|
| McDonald's | $1.25 | $3.00 |
| Chick-fil-A | $5.25 | $3.50 |
Matrix Representation
\(A =\)
1.25 3.00 5.25 3.50
← Element
Price of Fries at McDonald's
2 Dimensions of a Matrix
Order Matters: \(r \times c\)
The dimensions (or size) are written as: # of \(\times\) # of
Memory Trick: Rows first (\(\leftrightarrow\)), then Columns (\(\updownarrow\))
State the dimensions for each matrix below:
\(\begin{bmatrix} 1.25 & 3.00 \\ 5.25 & 3.50 \end{bmatrix}\)
Dim:
\(\begin{bmatrix} 5 & 3 & 7 \\ 2 & 1 & 4 \end{bmatrix}\)
Dim:
\(\begin{bmatrix} 7 & 6 & 5 \end{bmatrix}\)
Dim:
\(\begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix}\)
Dim:
3 Special Types of Matrices
Square Matrix \((r = c)\)
# of rows is to # of columns.
\(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) \(2 \times 2\)
Row Matrix \((1 \times c)\)
A matrix with exactly row.
\(\begin{bmatrix} 7 & 6 & 5 \end{bmatrix}\) \(1 \times 3\)
Column Matrix \((r \times 1)\)
A matrix with exactly column.
\(\begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix}\) \(3 \times 1\)
4 Element Notation & Interpretation
\(A_{rc}\) where \(r = \text{Row}\), \(c = \text{Col}\)
Given Matrix \(A\): \(3 \times 3\)
\(A = \begin{bmatrix} 5 & 3 & 7 \\ 2 & 1 & 4 \\ -3 & 0 & 2 \end{bmatrix}\)
\(A_{22} =\)
\(A_{32} =\)
\(A_{13} =\)
\(A_{31} =\)
Contextual Interpretation:
From the fast-food price matrix: \(A = \begin{bmatrix} 1.25 & 3.00 \\ 5.25 & 3.50 \end{bmatrix}\)
Interpret the element \(A_{11}\) in complete context:
Unit 3: Introduction to Matrices Page 1 of 2
Matrix Addition & Subtraction
Guided Lecture Notes • Operations & Dimensional Rules
Part 2 of 2
Fundamental Rule of Matrix Addition & Subtraction
Matrices can ONLY be added or subtracted if they have the EXACT SAME !
• Computation: Add or subtract corresponding (elements in matching positions).
• Mismatched dimensions: Operation is NOT POSSIBLE (undefined).
5 Guided Practice: \(2 \times 2\) Matrices
\(A = \begin{bmatrix} 5 & 3 \\ -2 & 0 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 7 \\ 3 & 4 \end{bmatrix}\)
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Find \(A + B\) Show work and final matrix:
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Find \(A - B\) Show work and final matrix:
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Find \(B - A\) Show work and final matrix:
Stop & Think: Does \(A - B = B - A\)? YES / NO
Why?
6 Different Dimensions & Feasibility Check
Always verify dimensions first!
MATRIX A \(\begin{bmatrix} 5 & 2 & 4 \\ 3 & -1 & -3 \end{bmatrix}\)
MATRIX B \(\begin{bmatrix} -2 & 6 & 1 \\ 0 & -3 & 4 \end{bmatrix}\)
MATRIX C \(\begin{bmatrix} -3 & -4 \\ 5 & 2 \\ 8 & 1 \end{bmatrix}\)
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Find \(A - B\) Show work and final matrix:
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Find \(B + A\) Show work and final matrix:
-
Find \(C + B\)
Dim \(C\):
Dim \(B\):
Result & Explanation:
Quick Check: Can a \(3 \times 4\) matrix be added to a \(4 \times 3\) matrix?
[ YES / NO ] Reason:
Unit 3: Matrix Operations Page 2 of 2
Matrix Foundations Answer Key
Matrix Foundations
Answer Key
Teacher Reference & Guided Solutions • Unit 3
Algebra II / Precalculus Page 1 of 2
1 What is a Matrix?
Key Definitions
A matrix (plural: matrices) is a rectangular arrangement of numbers organized in rows and columns used to collect and display data.
Each individual value in a matrix is called an element (also known as a component).
Real-World Data Table
| Store | Chicken | Fries |
|---|
| McDonald's | $1.25 | $3.00 |
| Chick-fil-A | $5.25 | $3.50 |
Matrix Representation
\(A =\)
1.25 3.00 5.25 3.50
Element \(A_{12}\)
McDonald's Fries Price
2 Dimensions of a Matrix
Order: \(r \times c\)
The dimensions are written as: # of Rows \(\times\) # of Columns
"RC Cola" : Horizontal (\(\leftrightarrow\)) by Vertical (\(\updownarrow\))
Dimensions Solutions:
\(\begin{bmatrix} 1.25 & 3.00 \\ 5.25 & 3.50 \end{bmatrix}\)
\(2 \times 2\)
\(\begin{bmatrix} 5 & 3 & 7 \\ 2 & 1 & 4 \end{bmatrix}\)
\(2 \times 3\)
\(\begin{bmatrix} 7 & 6 & 5 \end{bmatrix}\)
\(1 \times 3\)
\(\begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix}\)
\(3 \times 1\)
3 Special Types of Matrices
Square Matrix \((r = c)\)
# of rows is equal to # of columns.
\(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) \(2 \times 2\)
Row Matrix \((1 \times c)\)
A matrix with exactly one row.
\(\begin{bmatrix} 7 & 6 & 5 \end{bmatrix}\) \(1 \times 3\)
Column Matrix \((r \times 1)\)
A matrix with exactly one column.
\(\begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix}\) \(3 \times 1\)
4 Element Notation & Interpretation
\(A_{rc}\) \(r = \text{Row (down)}\), \(c = \text{Column (across)}\)
Given Matrix \(A\) (\(3 \times 3\)):
\(A = \begin{bmatrix} 5 & 3 & 7 \\ 2 & 1 & 4 \\ -3 & 0 & 2 \end{bmatrix}\)
\(A_{22} =\) 1
\(A_{32} =\) 0
\(A_{13} =\) 7
\(A_{31} =\) -3
