Determinant of a Matrix | Moosa Academy

Determinant of a Matrix - 2×2 Matrices

1️⃣ Understand what a determinant is and when it can be calculated

2️⃣ Learn the requirement for square matrices

3️⃣ Master the 2×2 determinant formula

4️⃣ Identify main diagonal and anti-diagonal elements

5️⃣ Practice with numerical examples step by step

📐 What is a Determinant?

A determinant is a special number that can be calculated from a square matrix. It's a fundamental concept in linear algebra with many applications, including solving systems of equations and finding matrix inverses.

Key Requirement: Square Matrix

To calculate a determinant, the matrix must be square
This means: number of rows = number of columns
Examples: 2×2, 3×3, 4×4 matrices
Cannot calculate determinant of 2×3 or 3×2 matrices

🔢 The 2×2 Determinant Formula

Let's start with the simplest square matrix - a 2×2 matrix. The formula is straightforward and easy to remember.

2×2 Determinant Formula

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Main diagonal × Main diagonal - Anti-diagonal × Anti-diagonal

🎯 Understanding the Diagonals

The key to calculating determinants is identifying the two diagonals correctly.

Diagonal Identification

Main Diagonal ↘️

\begin{vmatrix} \color{red}{a} & b \\ c & \color{red}{d} \end{vmatrix}

Elements: a and d

Top-left to bottom-right

Anti-Diagonal ↙️

\begin{vmatrix} a & \color{blue}{b} \\ \color{blue}{c} & d \end{vmatrix}

Elements: b and c

Top-right to bottom-left

📝 Example 1: Step-by-Step Calculation

Let's work through the example from the audio: calculating the determinant of a matrix with first row [3, 4] and second row [2, 5].

Example 1 Solution

Given Matrix:

\begin{vmatrix} 3 & 4 \\ 2 & 5 \end{vmatrix}

Step 1: Identify the diagonals

Main diagonal: 3 × 5

Anti-diagonal: 2 × 4

Step 2: Apply the formula

Determinant = (3 × 5) - (2 × 4)

Step 3: Calculate

= 15 - 8 = 7

📝 Example 2: Working with Negative Numbers

Let's work through the second example: a matrix with first row [-1, 2] and second row [3, 0]. Pay attention to the negative sign!

Example 2: Negative Numbers

Given Matrix:

\begin{vmatrix} -1 & 2 \\ 3 & 0 \end{vmatrix}

Step 1: Identify elements

a = -1, b = 2, c = 3, d = 0

Step 2: Apply formula carefully

Determinant = ad - bc

= (-1)(0) - (2)(3)

= 0 - 6 = -6

⚡ Quick Method: Remember the Pattern

Quick Pattern

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a \times d - b \times c

|c d| ↘️ ↙️

Memory tip: "Down-right MINUS up-right"

🧠 Key Points to Remember

Essential Rules

Square matrices only - rows = columns
Formula: ad - bc (not ad + bc)
Main diagonal first - then subtract anti-diagonal
Watch the signs - especially with negative numbers
Order matters - always subtract in the right direction
The minus sign is crucial - don't forget it!

🎯 Common Mistakes to Avoid

Forgetting the minus sign: It's ad - bc, not ad + bc
Wrong diagonal identification: Make sure you identify elements correctly
Sign errors: Be extra careful with negative numbers
Non-square matrices: Remember, determinants only exist for square matrices
Calculation errors: Double-check your arithmetic