Determinant of a 3×3 Matrix Calculator

Published on: November 2 2025
Final Answer: Free • Full Steps: Pro

This Determinant of a 3×3 Matrix Calculator finds the determinant of a \(3 \times 3\) matrix and shows the working line by line. Each entry of the top row is multiplied by the determinant of the \(2 \times 2\) block left when its own row and column are covered, and the middle part is subtracted rather than added. That alternating sign is the step most often missed. Following the same pattern every time makes the determinant easy to check and repeat.

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Work out each of the three parts.
  4. Combine the three parts.

Formula:

\(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\)
See Example 1 Hide Example 1

Problem

\(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Work out each of the three parts.
  4. Combine the three parts.

Step 1

Step 1 - Set up the problem.

In this problem: Labelling the entries left to right and top to bottom gives \(a\) through \(i\).

\(\begin{gathered} a = 1,\; b = 2,\; c = 3 \\ d = 4,\; e = 5,\; f = 6 \\ g = 7,\; h = 8,\; i = 10 \end{gathered}\)

Step 2

Step 2 - Write the formula.

In this problem: Each entry of the top row is multiplied by the determinant of the 2 by 2 block left when its own row and column are covered, and the middle one is subtracted.

\(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\)

Step 3

Step 3 - Work out each of the three parts.

In this problem: Three parts are still written as expressions, so each one is worked out in turn.

\(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\)

Step 3a

Step 3a - Work out each of the three parts.

In this problem: Inside the brackets \(5 \cdot 10 - 6 \cdot 8 = 2\), and \(1 \cdot 2 = 2\).

\(a(ei - fh) = 1(5 \cdot 10 - 6 \cdot 8) = 1 \cdot 2 = 2\)

Step 3b

Step 3b - Work out each of the three parts.

In this problem: Inside the brackets \(4 \cdot 10 - 6 \cdot 7 = -2\), and \(2 \cdot (-2) = -4\).

\(b(di - fg) = 2(4 \cdot 10 - 6 \cdot 7) = 2 \cdot (-2) = -4\)

Step 3c

Step 3c - Work out each of the three parts.

In this problem: Inside the brackets \(4 \cdot 8 - 5 \cdot 7 = -3\), and \(3 \cdot (-3) = -9\).

\(c(dh - eg) = 3(4 \cdot 8 - 5 \cdot 7) = 3 \cdot (-3) = -9\)

Step 4

Step 4 - Combine the three parts.

In this problem: The middle part is subtracted and the other two are added, so \(2 - (-4) + (-9) = -3\).

\(\det(A) = 2 - (-4) + (-9) = -3\)

Final Answer

\(\det(A) = -3\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula.
  3. Work out each of the three parts.
  4. Combine the three parts.

Step 1

Step 1 - Set up the problem.

In this problem: Labelling the entries left to right and top to bottom gives \(a\) through \(i\).

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

Step 2

Step 2 - Write the formula.

In this problem: Each entry of the top row is multiplied by the determinant of the 2 by 2 block left when its own row and column are covered, and the middle one is subtracted.

\(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\)

Step 3

Step 3 - Work out each of the three parts.

In this problem: Three parts are still written as expressions, so each one is worked out in turn.

\(\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\)

Step 3a

Step 3a - Work out each of the three parts.

In this problem: Inside the brackets \((-1) \cdot 5 - 2 \cdot 4 = -13\), and \(2 \cdot (-13) = -26\).

\(a(ei - fh) = 2((-1) \cdot 5 - 2 \cdot 4) = 2 \cdot (-13) = -26\)

Step 3b

Step 3b - Work out each of the three parts.

In this problem: Inside the brackets \(3 \cdot 5 - 2 \cdot 1 = 13\), and \(0 \cdot 13 = 0\).

\(b(di - fg) = 0(3 \cdot 5 - 2 \cdot 1) = 0 \cdot 13 = 0\)

Step 3c

Step 3c - Work out each of the three parts.

In this problem: Inside the brackets \(3 \cdot 4 - (-1) \cdot 1 = 13\), and \(1 \cdot 13 = 13\).

\(c(dh - eg) = 1(3 \cdot 4 - (-1) \cdot 1) = 1 \cdot 13 = 13\)

Step 4

Step 4 - Combine the three parts.

In this problem: The middle part is subtracted and the other two are added, so \(-26 - 0 + 13 = -13\).

\(\det(A) = -26 - 0 + 13 = -13\)

Final Answer

\(\det(A) = -13\)
Matrix A

Example Problems

Click one to load it into the box and solve it.