Determinant of a 3x3 Matrix Example

3×3 Matrix Determinants - Cofactor Expansion

1️⃣ Understand cofactor expansion along the first row

2️⃣ Learn to identify and "cover" rows and columns

3️⃣ Master the alternating sign pattern (+, -, +)

4️⃣ Calculate 2×2 determinants from the remaining elements

5️⃣ Practice with complete worked examples

📐 3×3 Determinant Overview

Calculating the determinant of a 3×3 matrix is more complex than a 2×2 matrix, but follows a systematic pattern. We use cofactor expansion along the first row, breaking it down into three 2×2 determinants.

3×3 Matrix General Form

\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}

We'll expand along the first row: a, b, c

🎯 The Cofactor Expansion Method

The method involves three steps for each element in the first row: take the element, cover its row and column, then multiply by the remaining 2×2 determinant. The key is the alternating sign pattern!

Key Pattern: Alternating Signs

First element (a): Positive (+)
Second element (b): Negative (-)
Third element (c): Positive (+)
Pattern: + - + - + - ...

📝 Step-by-Step Process

The Three-Step Process

Step 1: First Element (Positive)

1. Take the first element (a) from the first row

2. Cover its row (row 1) and column (column 1)

3. Multiply by the remaining 2×2 determinant

4. Sign: Positive (+)

Step 2: Second Element (Negative)

1. Take the second element (b) from the first row

2. Cover its row (row 1) and column (column 2)

3. Multiply by the remaining 2×2 determinant

4. Sign: Negative (-) - Very important!

Step 3: Third Element (Positive)

1. Take the third element (c) from the first row

2. Cover its row (row 1) and column (column 3)

3. Multiply by the remaining 2×2 determinant

4. Sign: Positive (+)

Complete Formula

\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - b\begin{vmatrix} d & f \\ g & i \end{vmatrix} + c\begin{vmatrix} d & e \\ g & h \end{vmatrix}

Notice the alternating signs: + - +

🔍 Visual Guide: Covering Rows and Columns

Let's visualize how to "cover" rows and columns to find the remaining 2×2 matrices for each step.

Covering Process Visualization

For element 'a':

\begin{vmatrix} \cancel{a} & \cancel{b} & \cancel{c} \\ \cancel{d} & e & f \\ \cancel{g} & h & i \end{vmatrix}

Remaining:

\begin{vmatrix} e & f \\ h & i \end{vmatrix}

For element 'b':

\begin{vmatrix} \cancel{a} & \cancel{b} & \cancel{c} \\ d & \cancel{e} & f \\ g & \cancel{h} & i \end{vmatrix}

Remaining:

\begin{vmatrix} d & f \\ g & i \end{vmatrix}

For element 'c':

\begin{vmatrix} \cancel{a} & \cancel{b} & \cancel{c} \\ d & e & \cancel{f} \\ g & h & \cancel{i} \end{vmatrix}

Remaining:

\begin{vmatrix} d & e \\ g & h \end{vmatrix}

🧮 Complete Worked Example

Let's work through a complete numerical example to see the method in action.

Example Problem

Calculate the determinant:

\begin{vmatrix} 2 & 1 & 3 \\ 0 & 4 & 1 \\ 5 & 2 & 1 \end{vmatrix}

Step 1: First element (2) - Positive

$+2 \times \begin{vmatrix} 4 & 1 \\ 2 & 1 \end{vmatrix} = +2 \times (4 \times 1 - 1 \times 2) = +2 \times 2 = +4$

Step 2: Second element (1) - Negative

$-1 \times \begin{vmatrix} 0 & 1 \\ 5 & 1 \end{vmatrix} = -1 \times (0 \times 1 - 1 \times 5) = -1 \times (-5) = +5$

Step 3: Third element (3) - Positive

$+3 \times \begin{vmatrix} 0 & 4 \\ 5 & 2 \end{vmatrix} = +3 \times (0 \times 2 - 4 \times 5) = +3 \times (-20) = -60$

Final Answer:

4 + 5 + (-60) = -51

🧠 Essential Points to Remember

Critical Rules

Alternating signs: + - + (first row expansion)
Cover completely: Remove entire row and column
Order matters: Go left to right along first row
2×2 determinants: Use ad - bc for each remaining matrix
Sign is crucial: Don't forget the negative for the middle term!
Final step: Add all three results together

⚠️ Common Mistakes to Avoid

Wrong signs: Remember the pattern is + - + for the first row
Incomplete covering: Make sure to cross out the entire row AND column
2×2 errors: Double-check each 2×2 determinant calculation
Addition mistakes: Be careful when combining the three terms
Element selection: Always use the first row elements in order