📐 Concept diagram

03-10 - Matrices (Introduction)

Phase: 3 | Subject: 03-10 Prerequisites: 01-09-polynomials.md (algebraic manipulation), 02-10-vectors-basic.md (vectors) Next subject: 04-01-limits.md

Learning Objectives

By the end of this subject, you will be able to:

1. Understand matrix notation and dimensions
2. Perform matrix addition, subtraction, and scalar multiplication
3. Multiply two matrices
4. Identify identity and zero matrices
5. Calculate 2×2 and 3×3 determinants
6. Find the inverse of a 2×2 matrix
7. Solve linear systems using matrix inverses

Core Content

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

Notation: Capital letters (A, B, C) for matrices. Entry aᵢⱼ means row i, column j.

Dimensions: m × n means m rows, n columns.

Example:

$A = [1 2 3]   is a 1×3 matrix (row vector)
    [4 5 6]       2×3 matrix
$

Matrix Addition and Subtraction

Only possible when matrices have the SAME dimensions.

Add/subtract corresponding entries:

$A + B = [a₁₁+b₁₁  a₁₂+b₁₂]
        [a₂₁+b₂₁  a₂₂+b₂₂]
$

Properties: - Commutative: A + B = B + A - Associative: (A + B) + C = A + (B + C)

Scalar Multiplication

Multiply every entry by the scalar:

3A = [3a₁₁ 3a₁₂] [3a₂₁ 3a₂₂]

Matrix Multiplication

Rule: A (m×n) × B (n×p) = C (m×p)

The number of columns of A must equal the number of rows of B.

Entry cᵢⱼ = sum of (row i of A) × (column j of B)

Example:

$A = [1 2]   B = [5 6]
    [3 4]       [7 8]

A × B = [1(5)+2(7)   1(6)+2(8)]  =  [19 22]
        [3(5)+4(7)   3(6)+4(8)]     [43 50]
$

Properties: - NOT commutative: AB ≠ BA in general - Associative: (AB)C = A(BC) - Distributive: A(B + C) = AB + AC

Identity Matrix

Iₙ (n×n) has 1s on the main diagonal and 0s elsewhere.

$I₂ = [1 0]
     [0 1]
$

Property: AI = IA = A for any square matrix A.

Zero Matrix

All entries are 0.

Property: A + 0 = 0 + A = A

Determinants (2×2 and 3×3)

2×2

$|a  b|
|c  d|  = ad - bc
$

Example: |1 2| = 1(5) - 2(3) = 5 - 6 = -1 |3 5|

3×3 (Sarrus' Rule or cofactor expansion)

Using cofactor expansion along the first row:

$|a b c|
|d e f|  = a(ei − fh) − b(di − fg) + c(dh − eg)
|g h i|
$

Example:

|1 2 3|
|0 1 4|  = 1(1·5 - 4·2) - 2(0·5 - 4·3) + 3(0·2 - 1·3)
|3 2 5|  = 1(5-8) - 2(0-12) + 3(0-3)
         = -3 + 24 - 9
         = 12

## Inverse of a 2×2 Matrix

For A = [a b; c d], if det(A) = ad - bc ≠ 0:

A⁻¹ = (1/det(A)) [ d -b] [-c a]

**Example:** A = [2 3; 1 4]
det(A) = 2(4) - 3(1) = 8 - 3 = 5
A⁻¹ = (1/5) [4 -3; -1 2] = [4/5  -3/5; -1/5  2/5]

**Verification:** A × A⁻¹ should equal I. Check: [2 3; 1 4] × [4/5 -3/5; -1/5 2/5] = [2(4/5)+3(-1/5) 2(-3/5)+3(2/5); 1(4/5)+4(-1/5) 1(-3/5)+4(2/5)] = [5/5 0; 0 5/5] = [1 0; 0 1]. ✓

## Solving Linear Systems with Matrices

A system of linear equations can be written as AX = B.

**Example:** 2x + 3y = 5, x + 4y = 6

Matrix form: [2 3; 1 4][x; y] = [5; 6]

Solution: X = A⁻¹B

A⁻¹ = [4/5 -3/5; -1/5 2/5]
X = [4/5 -3/5; -1/5 2/5] × [5; 6] = [(4/5)(5)+(-3/5)(6); (-1/5)(5)+(2/5)(6)] = [4-3.6; -1+2.4] = [0.4; 1.4]

So x = 0.4 = 2/5, y = 1.4 = 7/5.

Key Terms

• 03 10 Matrices Introduction
• 3×3 (Sarrus' Rule or cofactor expansion)
• Correct: A)
• Correct: C)
• Determinants (2×2 and 3×3)
• Entry cᵢⱼ = sum of (row i of A) × (column j of B)
• Example 1: Matrix multiplication
• Example 2: Determinant
• Identity Matrix
• Inverse of a 2×2 Matrix
• Matrix Addition and Subtraction
• Matrix Multiplication

Worked Examples

Example 1: Matrix multiplication

$A = [1 -1]   B = [2  3]
    [0  2]       [1  0]

A × B = [1(2)+(-1)(1)   1(3)+(-1)(0)]  =  [1  3]
        [0(2)+2(1)      0(3)+2(0)]       [2  0]
$

Example 2: Determinant

|2 3 1| |4 1 2| = 2(1·5 - 2·3) - 3(4·5 - 2·7) + 1(4·3 - 1·7) |7 3 5| = 2(5 - 6) - 3(20 - 14) + 1(12 - 7) = 2(-1) - 3(6) + 5 = -2 - 18 + 5 = -15

Quiz

Q1: What does the concept of Identity Matrix primarily refer to in this subject?

A) A visual representation of Identity Matrix B) A computational error related to Identity Matrix C) A historical anecdote about Identity Matrix D) The definition and application of Identity Matrix

Correct: D)

• If you chose A: This is incorrect. Identity Matrix is defined as: the definition and application of identity matrix. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Identity Matrix is defined as: the definition and application of identity matrix. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Identity Matrix is defined as: the definition and application of identity matrix. The other options describe different aspects that are not the primary focus.
• If you chose D: Identity Matrix is defined as: the definition and application of identity matrix. The other options describe different aspects that are not the primary focus. Correct!

Q2: What is the primary purpose of Matrix Addition and Subtraction?

A) It is primarily a historical notation system B) It replaces all other methods in this domain C) It is used only in advanced research contexts D) It is used to matrix addition and subtraction in mathematical analysis

Correct: D)

• If you chose A: This is incorrect. Matrix Addition and Subtraction serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Matrix Addition and Subtraction serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Matrix Addition and Subtraction serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: Matrix Addition and Subtraction serves the purpose described in the correct answer. The other options misrepresent its role. Correct!

Q3: Which statement about Matrix Multiplication is TRUE?

A) Matrix Multiplication is mentioned only as a historical footnote B) Matrix Multiplication is an advanced topic beyond this subject's scope C) Matrix Multiplication is a fundamental concept covered in this subject D) Matrix Multiplication is not related to this subject

Correct: C)

• If you chose A: This is incorrect. Matrix Multiplication is a fundamental concept covered in this subject. This subject covers Matrix Multiplication as part of its core content.
• If you chose B: This is incorrect. Matrix Multiplication is a fundamental concept covered in this subject. This subject covers Matrix Multiplication as part of its core content.
• If you chose C: Matrix Multiplication is a fundamental concept covered in this subject. This subject covers Matrix Multiplication as part of its core content. Correct!
• If you chose D: This is incorrect. Matrix Multiplication is a fundamental concept covered in this subject. This subject covers Matrix Multiplication as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) An unrelated numerical value B) ** [4 6] / [12 14] C) A different result from a common mistake D) The inverse of the correct answer

Correct: B)

• If you chose A: This is incorrect. The worked examples show that the result is ** [4 6] / [12 14]. The other options represent common errors.
• If you chose B: The worked examples show that the result is ** [4 6] / [12 14]. The other options represent common errors. Correct!
• If you chose C: This is incorrect. The worked examples show that the result is ** [4 6] / [12 14]. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is ** [4 6] / [12 14]. The other options represent common errors.

Q5: How are Matrix Multiplication and What Is A Matrix? related?

A) Matrix Multiplication and What Is A Matrix? are closely related concepts B) Matrix Multiplication is the inverse of What Is A Matrix? C) Matrix Multiplication is a special case of What Is A Matrix? D) Matrix Multiplication and What Is A Matrix? are completely unrelated topics

Correct: A)

• If you chose A: Both Matrix Multiplication and What Is A Matrix? are covered in this subject as interconnected topics. Correct!
• If you chose B: This is incorrect. Both Matrix Multiplication and What Is A Matrix? are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Matrix Multiplication and What Is A Matrix? are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Matrix Multiplication and What Is A Matrix? are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Scalar Multiplication?

A) Scalar Multiplication has no common misconceptions B) Scalar Multiplication is always computed the same way in all contexts C) A common mistake is confusing Scalar Multiplication with a similar concept D) The main error with Scalar Multiplication is using it when it is not needed

Correct: C)

• If you chose A: This is incorrect. Students often confuse Scalar Multiplication with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Scalar Multiplication with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: Students often confuse Scalar Multiplication with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose D: This is incorrect. Students often confuse Scalar Multiplication with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Zero Matrix?

A) Zero Matrix is not practically useful B) Apply Zero Matrix to solve problems in this subject's domain C) Avoid Zero Matrix unless explicitly instructed D) Use Zero Matrix only in pure mathematics contexts

Correct: B)

• If you chose A: This is incorrect. Zero Matrix is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: Zero Matrix is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. Zero Matrix is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. Zero Matrix is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Add: [1 2] + [3 4] [5 6] [7 8] Answer: [4 6] / [12 14]

2. Scalar: 2 × [1 -2] [3 4] Answer: [2 -4] / [6 8]

3. Multiply: [1 2] × [1 0] [3 4] [0 1] Answer: [1 2] / [3 4]

4. Determinant: |1 2| / |3 4| Answer: 1(4) - 2(3) = 4 - 6 = -2

5. Determinant: |2 0 1| / |1 3 2| / |4 1 5| Answer: 2(15-2) - 0 + 1(1-12) = 26 - 11 = 15

6. Inverse of [3 1; 5 2] Answer: det = 3(2) - 1(5) = 1. A⁻¹ = [2 -1; -5 3].

7. Solve using matrices: 3x + y = 7, 5x + 2y = 12 Answer: A = [3 1; 5 2], det = 1, A⁻¹ = [2 -1; -5 3]. X = [2 -1; -5 3] × [7; 12] = [14-12; -35+36] = [2; 1]. So x = 2, y = 1.

Summary

Key takeaways:

• Matrix: m×n array of numbers
• Addition: same dimensions, add corresponding entries
• Scalar: multiply every entry
• Multiplication: (m×n) × (n×p) = (m×p), sum of products
• Multiplication is NOT commutative
• Identity Iₙ: 1s on diagonal, 0s elsewhere
• 2×2 determinant: ad - bc
• 3×3 determinant: cofactor expansion
• 2×2 inverse: (1/det)[d -b; -c a] (when det ≠ 0)
• Linear systems: AX = B → X = A⁻¹B

Pitfalls

• Multiplying matrices with incompatible dimensions: To multiply A × B, the number of columns in A must equal the number of rows in B. For a 2×3 and a 4×2, the product is undefined. Always write the dimensions and check that the inner dimensions match: (m×n) × (n×p) = (m×p).
• Assuming matrix multiplication is commutative: In general, AB ≠ BA. Even when both products are defined (e.g., both matrices are square), they usually differ. Treat matrix multiplication as non-commutative by default — only special pairs (e.g., diagonal matrices, identity) commute.
• Sign errors in 2×2 determinant and inverse: The determinant is ad - bc, not bc - ad. The inverse of [a b; c d] swaps a and d (with d keeping its sign and a keeping its sign) and negates b and c. A common error is negating the wrong entries or getting the sign on a and d wrong.
• Forgetting to check det(A) ≠ 0 before inverting: A matrix is invertible only when its determinant is non-zero. If det(A) = 0, A is singular and has no inverse — attempting to apply the inverse formula gives division by zero. Always compute the determinant first.
• Adding matrices of different dimensions: Matrix addition requires BOTH matrices to have exactly the same number of rows AND columns. Adding a 2×3 to a 3×2 is undefined — you can't add corresponding entries when the matrices have different shapes.

Next Steps

Next up: 04-01-limits.md