Determinant of a 3×3 Matrix Calculator

This Determinant of a 3×3 Matrix Calculator helps you find the determinant of a 3×3 matrix and shows each step clearly. It works by expanding the matrix using cofactors from the first row, then simplifying the expression into one determinant value. This makes it useful for checking answers, understanding how a determinant is calculated, and practising linear algebra step by step.

Step-by-step method

1. Identify the entries a through i in the 3×3 matrix.
2. Find the determinant by breaking it into the a-part, b-part, and c-part.
3. Combine the parts using det( A ) = a-part − b-part + c-part.

Formula:

Example 1: Take the values below.

Step 1 - Identify the entries a through i in the 3×3 matrix.

In this problem: For a 3×3 matrix, label the entries from left to right and top to bottom as a through i.

Step 2A - Find the a-part of the determinant.

In this problem: The a-part is \(a(ei - fh)\).

Step 2B - Find the b-part of the determinant.

In this problem: The b-part is \(b(di - fg)\). This part is subtracted in the determinant formula.

Step 2C - Find the c-part of the determinant.

In this problem: The c-part is \(c(dh - eg)\).

Step 2D - Combine the determinant parts.

In this problem: The determinant is \(-306\).

Final answer:

Example 2: Take the values below.

Step 1 - Identify the entries a through i in the 3×3 matrix.

In this problem: For a 3×3 matrix, label the entries from left to right and top to bottom as a through i.

Step 2A - Find the a-part of the determinant.

In this problem: The a-part is \(a(ei - fh)\).

Step 2B - Find the b-part of the determinant.

In this problem: The b-part is \(b(di - fg)\). This part is subtracted in the determinant formula.

Step 2C - Find the c-part of the determinant.

In this problem: The c-part is \(c(dh - eg)\).

Step 2D - Combine the determinant parts.

In this problem: The determinant is \(49\).

Final answer:

See Example 1 Hide Example 1

Problem

\(A = \begin{pmatrix} 6 & 1 & 1 \\ 4 & -2 & 5 \\ 2 & 8 & 7 \end{pmatrix}\)

Approach

Step-by-step method

1. Identify the entries a through i in the 3×3 matrix.
2. Find the determinant by breaking it into the a-part, b-part, and c-part.
3. Combine the parts using det( A ) = a-part − b-part + c-part.

Step 1

Step 1 - Identify the entries a through i in the 3×3 matrix.

In this problem: For a 3×3 matrix, label the entries from left to right and top to bottom as a through i.

\(\begin{gathered} a = 6,\; b = 1,\; c = 1 \\ d = 4,\; e = -2,\; f = 5 \\ g = 2,\; h = 8,\; i = 7 \end{gathered}\)

Step 2A

Step 2A - Find the a-part of the determinant.

In this problem: The a-part is \(a(ei - fh)\).

\(a(ei - fh) = 6((-2) \cdot 7 - 5 \cdot 8) = -324\)

Step 2B

Step 2B - Find the b-part of the determinant.

In this problem: The b-part is \(b(di - fg)\). This part is subtracted in the determinant formula.

\(b(di - fg) = 1(4 \cdot 7 - 5 \cdot 2) = 18\)

Step 2C

Step 2C - Find the c-part of the determinant.

In this problem: The c-part is \(c(dh - eg)\).

\(c(dh - eg) = 1(4 \cdot 8 - (-2) \cdot 2) = 36\)

Step 2D

Step 2D - Combine the determinant parts.

In this problem: The determinant is \(-306\).

\(\det(A) = -324 - 18 + 36 = -306\)

Final Answer

\(\det(A) = -306\)

See Example 2 Hide Example 2

Problem

\(A = \begin{pmatrix} 2 & -3 & 1 \\ 2 & 0 & -1 \\ 1 & 4 & 5 \end{pmatrix}\)

Approach

Step-by-step method

1. Identify the entries a through i in the 3×3 matrix.
2. Find the determinant by breaking it into the a-part, b-part, and c-part.
3. Combine the parts using det( A ) = a-part − b-part + c-part.

Step 1

Step 1 - Identify the entries a through i in the 3×3 matrix.

In this problem: For a 3×3 matrix, label the entries from left to right and top to bottom as a through i.

\(\begin{gathered} a = 2,\; b = -3,\; c = 1 \\ d = 2,\; e = 0,\; f = -1 \\ g = 1,\; h = 4,\; i = 5 \end{gathered}\)

Step 2A

Step 2A - Find the a-part of the determinant.

In this problem: The a-part is \(a(ei - fh)\).

\(a(ei - fh) = 2(0 \cdot 5 - (-1) \cdot 4) = 8\)

Step 2B

Step 2B - Find the b-part of the determinant.

In this problem: The b-part is \(b(di - fg)\). This part is subtracted in the determinant formula.

\(b(di - fg) = (-3)(2 \cdot 5 - (-1) \cdot 1) = -33\)

Step 2C

Step 2C - Find the c-part of the determinant.

In this problem: The c-part is \(c(dh - eg)\).

\(c(dh - eg) = 1(2 \cdot 4 - 0 \cdot 1) = 8\)

Step 2D

Step 2D - Combine the determinant parts.

In this problem: The determinant is \(49\).

\(\det(A) = 8 - (-33) + 8 = 49\)

Final Answer

\(\det(A) = 49\)