Determinant of Matrices 3x3

3×3 Matrix Determinant - Cofactor Expansion Method

1️⃣ Learn the cofactor expansion method for 3×3 determinants

2️⃣ Understand the alternating sign pattern (+, -, +)

3️⃣ Master creating 2×2 submatrices by covering rows and columns

4️⃣ Practice calculating multiple 2×2 determinants systematically

5️⃣ Combine all results to get the final 3×3 determinant

🎯 The 3×3 Determinant Method

To calculate a 3×3 determinant, we expand along the first row. We take each element from the first row, create a 2×2 submatrix, and use the alternating sign pattern.

3×3 Determinant Formula

\det\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} = a_{11}\begin{vmatrix} \cdot & \cdot \\ \cdot & \cdot \end{vmatrix} - a_{12}\begin{vmatrix} \cdot & \cdot \\ \cdot & \cdot \end{vmatrix} + a_{13}\begin{vmatrix} \cdot & \cdot \\ \cdot & \cdot \end{vmatrix}

Expand along first row with alternating signs: +, -, +

📋 Step 1: Set Up the Expansion Framework

First, we write down each element from the first row, create empty 2×2 determinants, and set up the alternating sign pattern. The negative sign is very important!

Framework Setup Process

Step 1: Take first element (a₁₁)

a_{11} \times \begin{vmatrix} \_ & \_ \\ \_ & \_ \end{vmatrix}

Write the first element and create an empty 2×2 determinant (positive)

Step 2: Take second element (a₁₂) with negative sign

- a_{12} \times \begin{vmatrix} \_ & \_ \\ \_ & \_ \end{vmatrix}

Add negative sign, then second element with empty 2×2 determinant

Step 3: Take third element (a₁₃) with positive sign

+ a_{13} \times \begin{vmatrix} \_ & \_ \\ \_ & \_ \end{vmatrix}

Add positive sign, then third element with empty 2×2 determinant

Framework:

a_{11}\begin{vmatrix} \_ & \_ \\ \_ & \_ \end{vmatrix} - a_{12}\begin{vmatrix} \_ & \_ \\ \_ & \_ \end{vmatrix} + a_{13}\begin{vmatrix} \_ & \_ \\ \_ & \_ \end{vmatrix}

🔍 Step 2: Fill the 2×2 Submatrices

For each 2×2 determinant, we cover the first row and the corresponding column, then write the remaining 4 elements in the 2×2 matrix.

Submatrix Creation Process

First Submatrix: Cover Row 1 and Column 1

\begin{bmatrix} \cancel{a_{11}} & \cancel{a_{12}} & \cancel{a_{13}} \\ \cancel{a_{21}} & a_{22} & a_{23} \\ \cancel{a_{31}} & a_{32} & a_{33} \end{bmatrix} \rightarrow \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix}

Cover first row and first column, take remaining elements

Second Submatrix: Cover Row 1 and Column 2

\begin{bmatrix} \cancel{a_{11}} & \cancel{a_{12}} & \cancel{a_{13}} \\ a_{21} & \cancel{a_{22}} & a_{23} \\ a_{31} & \cancel{a_{32}} & a_{33} \end{bmatrix} \rightarrow \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix}

Cover first row and second column, take remaining elements

Third Submatrix: Cover Row 1 and Column 3

\begin{bmatrix} \cancel{a_{11}} & \cancel{a_{12}} & \cancel{a_{13}} \\ a_{21} & a_{22} & \cancel{a_{23}} \\ a_{31} & a_{32} & \cancel{a_{33}} \end{bmatrix} \rightarrow \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

Cover first row and third column, take remaining elements

🧮 Step 3: Calculate Each 2×2 Determinant

Now we calculate each 2×2 determinant using the formula we learned: ad - bc. Then multiply by the corresponding first row element and apply the correct sign.