Lesson 7: Matrices and Determinants

👨🏽‍🏫 Introduction

Imagine organizing student test scores, traffic data, or even transforming images with computers — all of these can be done using matrices. A matrix is a simple but powerful tool for storing and working with data in rows and columns. In this lesson, we’ll explore what matrices are, how to operate on them, and how to find the determinant of a 2×2 matrix.

By the end of this lesson, you will:

• Understand what matrices are and their types
• Add, subtract, and multiply matrices
• Use scalar multiplication
• Find the determinant of a 2×2 matrix

🧠 Key Concepts

📌 1. What Is a Matrix?

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

Example:

This is a 2×2 matrix (2 rows, 2 columns).

📐 2. Types of Matrices

• Row matrix: Only one row → e.g.
• Column matrix: Only one column → e.g.
• Square matrix: Same number of rows and columns
• Zero matrix: All elements are 0 →
• Identity matrix: 1s on the diagonal, 0s elsewhere →
  

➕ 3. Matrix Addition and Subtraction

You can only add or subtract matrices of the same order (same size).

Example:

✖️ 4. Scalar Multiplication

Multiply each element in the matrix by a number (called a scalar).

Example:

🔢 5. Matrix Multiplication

To multiply A × B, the number of columns in A must equal the number of rows in B.

Example:

Note: Matrix multiplication is not commutative (i.e., AB ≠ BA)

🧮 6. Determinant of a 2×2 Matrix

For matrix , the determinant is:

Example:

🧮 Sample Problem Walkthroughs

1. Add:
   
2. Multiply: 2 ×
   
3. Find determinant:
   
   

✍🏽 Practice Exercises

🅰️ A. Add or Subtract

🅱️ B. Scalar and Matrix Multiplication

1. Multiply 3 ×
2. 

🅾️ C. Determinants

1. Find det of
2. Find det of

✅ Answers and Explanations

• A1:

• B1:
• B2:

• C1:
• C2:

📌 Recap

• ✅ Matrices are rectangular arrays used to represent and organize numbers
• ✅ You can add/subtract matrices of the same order
• ✅ Scalar multiplication multiplies every element by a number
• ✅ Matrix multiplication requires matching dimensions
• ✅ The determinant of a 2×2 matrix is

💭 Reflection Prompt

Think about how spreadsheets, traffic patterns, or even graphics in video games use grids. How might matrices help solve real-world problems?