Calculators / Linear Algebra / Determinant of a 3×3 Matrix Calculator

Determinant of a 3×3 Matrix Calculator

Published on: November 2 2025

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 or an equivalent determinant method to combine the entries correctly into a single 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. Write the 3×3 matrix a.
  2. Use the 3×3 determinant expansion formula.
  3. Substitute values and compute det(a).

Determinant formula

det(a) = a(e·i − f·h) − b(d·i − f·g) + c(d·h − e·g)
, where a =
a
b
c
d
e
f
g
h
i

Example 1: det(a) for a 3×3 matrix

Step 1 - Write the 3×3 matrix a.

In this problem: We start with the given 3×3 matrix.

a =
1
2
3
0
4
5
1
0
6

Step 2 - Use the 3×3 determinant expansion formula.

In this problem: Use the standard 3×3 expansion.

det(a) = a(e·i − f·h) − b(d·i − f·g) + c(d·h − e·g)
, where a =
a
b
c
d
e
f
g
h
i

Step 3 - Substitute values and compute det(a).

In this problem: Substitute the entries into the formula.

det(a) =
1·(4·6 − 5·0) − 2·(0·6 − 5·1) + 3·(0·0 − 4·1)

Step 4 - Simplify to get the final value.

In this problem: Compute the arithmetic.

det(a) =
1·(4·6 − 5·0) − 2·(0·6 − 5·1) + 3·(0·0 − 4·1) = 22

Final answer: 22

Example 2: det(a) for a 3×3 matrix

Step 1 - Write the 3×3 matrix a.

In this problem: We start with the given 3×3 matrix.

a =
-2
1
3
4
0
-1
2
5
1

Step 2 - Use the 3×3 determinant expansion formula.

In this problem: Use the standard 3×3 expansion.

det(a) = a(e·i − f·h) − b(d·i − f·g) + c(d·h − e·g)
, where a =
a
b
c
d
e
f
g
h
i

Step 3 - Substitute values and compute det(a).

In this problem: Substitute the entries into the formula.

det(a) =
( -2 )·(0·1 − ( -1 )·5) − 1·(4·1 − ( -1 )·2) + 3·(4·5 − 0·2)

Step 4 - Simplify to get the final value.

In this problem: Compute the arithmetic.

det(a) =
( -2 )·(0·1 − ( -1 )·5) − 1·(4·1 − ( -1 )·2) + 3·(4·5 − 0·2) = 44

Final answer: 44

Matrix a