Determinant of Matrices 2x2

2×2 Matrix Determinant - Formula and Calculation

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

2️⃣ Learn the requirement for square matrices (equal rows and columns)

3️⃣ Master the 2×2 determinant formula using diagonals

4️⃣ Practice calculating determinants with positive and negative elements

5️⃣ Handle negative signs correctly in diagonal calculations

📐 What is a Matrix Determinant?

To calculate the determinant of a matrix, the number of rows must equal the number of columns. This is called a square matrix because rows = columns.

Square Matrix Requirement

\text{For determinant calculation: } \text{Rows} = \text{Columns}

Only square matrices have determinants

Examples of Square Matrices:

2×2 matrix: 2 rows and 2 columns ✓
3×3 matrix: 3 rows and 3 columns ✓
4×4 matrix: 4 rows and 4 columns ✓
2×3 matrix: 2 rows but 3 columns ❌ (not square)

🎯 The 2×2 Determinant Formula

For a 2×2 matrix, we calculate the determinant by multiplying the main diagonal, then subtracting the secondary diagonal. The negative sign is very important!

2×2 Determinant Formula

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

Main diagonal minus secondary diagonal

Diagonal Identification

✅ Main Diagonal (Top-left to Bottom-right)

\begin{bmatrix} \color{green}{a} & b \\ c & \color{green}{d} \end{bmatrix} \rightarrow a \times d

Multiply the top-left and bottom-right elements

⚠️ Secondary Diagonal (Top-right to Bottom-left)

\begin{bmatrix} a & \color{red}{b} \\ \color{red}{c} & d \end{bmatrix} \rightarrow b \times c

Multiply the top-right and bottom-left elements (with negative sign)

🔢 Step-by-Step Example 1: Basic Calculation

Let's calculate the determinant of the matrix $\begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$ step by step.

Example 1:

\begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}

Step 1: Identify the elements

a = 2, b = 4, c = 1, d = 3

Step 2: Calculate main diagonal

Main diagonal:

a \times d = 2 \times 3 = 6

Step 3: Calculate secondary diagonal

Secondary diagonal:

b \times c = 4 \times 1 = 4

Step 4: Apply the formula

Determinant:

ad - bc = 6 - 4 = 2

Final Answer:

\det\begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} = 2

⚠️ Example 2: Handling Negative Numbers

The most challenging part is when the secondary diagonal has negative numbers. Let's see what happens when we have $\begin{bmatrix} 2 & -4 \\ 1 & 3 \end{bmatrix}$ .

Example 2:

\begin{bmatrix} 2 & -4 \\ 1 & 3 \end{bmatrix}

Step 1: Identify the elements

a = 2, b = -4, c = 1, d = 3

Step 2: Calculate main diagonal

Main diagonal:

a \times d = 2 \times 3 = 6

Step 3: Calculate secondary diagonal carefully

Secondary diagonal:

b \times c = (-4) \times 1 = -4

Step 4: Apply the formula (negative × negative = positive)

Determinant:

ad - bc = 6 - (-4) = 6 + 4 = 10

Note: The negative sign in the formula times the negative result gives a positive

Final Answer:

\det\begin{bmatrix} 2 & -4 \\ 1 & 3 \end{bmatrix} = 10

🧮 Key Points About Signs

Sign Rules for Determinants

🔹 The formula always has a negative sign

$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$ ← This negative is part of the formula

🔹 When secondary diagonal is negative

If $bc$ gives a negative result, then $-(negative) = positive$

Example: $6 - (-4) = 6 + 4 = 10$

🔹 The determinant is always a single number

The result is a constant (scalar), not a matrix

📝 Practice Examples

Quick Practice Problems

Example A

\det\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Main diagonal:

1 \times 4 = 4

Secondary diagonal:

2 \times 3 = 6

Result:

4 - 6 = -2

Example B

\det\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}

Main diagonal:

3 \times 5 = 15

Secondary diagonal:

1 \times 2 = 2

Result:

15 - 2 = 13

Example C

\det\begin{bmatrix} -1 & 3 \\ 2 & 4 \end{bmatrix}

Main diagonal:

(-1) \times 4 = -4

Secondary diagonal:

3 \times 2 = 6

Result:

-4 - 6 = -10

Example D

\det\begin{bmatrix} 2 & -1 \\ -3 & 4 \end{bmatrix}

Main diagonal:

2 \times 4 = 8

Secondary diagonal:

(-1) \times (-3) = 3

Result:

8 - 3 = 5

🧠 Determinant Calculation Summary

Step-by-Step Process

Check Square Matrix - Ensure rows = columns
Identify Elements - Label a, b, c, d positions
Calculate Main Diagonal - Multiply a × d
Calculate Secondary Diagonal - Multiply b × c
Apply Formula - ad - bc
Handle Signs Carefully - Watch for negative × negative = positive

🎯 Key Determinant Rules

Square Matrix Only: Determinants only exist for square matrices (n×n)
Formula for 2×2: $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$
Diagonal Method: Main diagonal minus secondary diagonal
Sign Importance: The negative sign in the formula is crucial
Negative Handling: When secondary diagonal is negative, minus negative = plus positive
Scalar Result: The determinant is always a single number, never a matrix