Introduction to Matrices

What is a Matrix?

Matrix: Rectangular array of numbers arranged in rows and columns

Notation: Usually capital letters (A, B, C)

Example:

A = [2  3]
    [4  5]
    [6  7]

Elements: Individual numbers in the matrix

Entry notation: aᵢⱼ = element in row i, column j

Matrix Dimensions

Dimension: rows × columns (read "m by n")

Written as: m × n matrix

Example dimensions:

[1  2  3]         2×3 matrix (2 rows, 3 columns)
[4  5  6]

[1  2]            3×2 matrix (3 rows, 2 columns)
[3  4]
[5  6]

[7]               3×1 matrix (column vector)
[8]
[9]

Example: Identify Dimensions

Matrix A:

[1  0  -2  5]
[3  1   4  0]
[6  2   1  3]

Dimensions: 3 × 4 (3 rows, 4 columns)

Special Types of Matrices

Square matrix: Same number of rows and columns (n × n)

Row matrix: 1 × n (single row)

Column matrix: m × 1 (single column)

Zero matrix: All entries are 0

Examples

Square matrix (2×2):

[3  1]
[4  2]

Row matrix (1×4):

[5  -1  0  3]

Column matrix (3×1):

[2]
[7]
[4]

Zero matrix (2×3):

[0  0  0]
[0  0  0]

Accessing Elements

Entry aᵢⱼ: Row i, column j

Example: Find Specific Entries

Matrix B:

[5   2  -1]
[3   0   4]
[-2  6   1]

Find b₁₂: Row 1, column 2 = 2

Find b₃₁: Row 3, column 1 = -2

Find b₂₃: Row 2, column 3 = 4

Matrix Equality

Two matrices are equal if:

1. Same dimensions
2. All corresponding entries are equal

Example: Equal Matrices

Are these equal?

A = [1  2]    B = [1  2]
    [3  4]        [3  4]

Yes: Same dimensions (2×2), all entries match

Not equal:

C = [1  2]    D = [1  2  0]
    [3  4]        [3  4  0]

Different dimensions: C is 2×2, D is 2×3

Adding Matrices

Add corresponding entries

Requirement: Matrices must have SAME dimensions

Formula: [A + B]ᵢⱼ = aᵢⱼ + bᵢⱼ

Example 1: Add Matrices

[1  2] + [5  1] = [1+5  2+1] = [6  3]
[3  4]   [2  3]   [3+2  4+3]   [5  7]

Example 2: Larger Matrices

[1   2  -1]   [4  0   2]   [5   2   1]
[0   3   5] + [1  2  -3] = [1   5   2]
[-2  4   0]   [0  1   1]   [-2  5   1]

Example 3: Cannot Add

[1  2] + [1  2  3]
[3  4]   [4  5  6]

Cannot add: Different dimensions (2×2 vs 2×3)

Subtracting Matrices

Subtract corresponding entries

Same requirement: Matrices must have same dimensions

Formula: [A - B]ᵢⱼ = aᵢⱼ - bᵢⱼ

Example 1: Subtract Matrices

[5  3] - [2  1] = [5-2  3-1] = [3  2]
[6  4]   [3  0]   [6-3  4-0]   [3  4]

Example 2: With Negatives

[4  -1]   [2   3]   [2  -4]
[0   5] - [-1  2] = [1   3]

Scalar Multiplication

Multiply every entry by a scalar (number)

Formula: [kA]ᵢⱼ = k · aᵢⱼ

Example 1: Multiply by Scalar

3 · [2  1] = [6  3]
    [4  0]   [12 0]

Example 2: Negative Scalar

-2 · [1  -3] = [-2   6]
     [5   0]   [-10  0]

Example 3: Fraction Scalar

1/2 · [4   6] = [2  3]
      [8  -2]   [4 -1]

Combining Operations

Use order of operations

Example 1: Add Then Multiply

Find: 2A + B where

A = [1  2]    B = [3  0]
    [3  4]        [1  2]

Calculate 2A:

2A = [2  4]
     [6  8]

Add B:

2A + B = [2  4] + [3  0] = [5  4]
         [6  8]   [1  2]   [7  10]

Example 2: Subtraction and Scalar

Find: 3A - 2B where

A = [4  1]    B = [2  3]
    [0  5]        [1  0]

Calculate 3A:

3A = [12  3]
     [0   15]

Calculate 2B:

2B = [4  6]
     [2  0]

Subtract:

3A - 2B = [12  3] - [4  6] = [8  -3]
          [0  15]   [2  0]   [-2  15]

Properties of Matrix Operations

Commutative (addition): A + B = B + A

Associative (addition): (A + B) + C = A + (B + C)

Distributive (scalar): k(A + B) = kA + kB

Additive identity: A + 0 = A (0 is zero matrix)

Example: Verify Commutative

A = [1  2]    B = [3  0]
    [4  5]        [1  2]

A + B = [4  2]
        [5  7]

B + A = [4  2]
        [5  7]

A + B = B + A ✓

Real-World Applications

Computer graphics: Transformations (rotate, scale, translate)

Economics: Input-output models

Network analysis: Connections between nodes

Data science: Organizing datasets

Physics: System of equations, quantum mechanics

Example: Student Scores

Matrix of test scores:

        Test1  Test2  Test3
Alice   [85    90     88]
Bob     [78    82     85]
Charlie [92    88     90]

Class averages as row matrix:

[85  86.7  87.7]

Practice