Determinant Calculator for 2×2, 3×3 and 4×4 Matrices with Steps
Type a square matrix one row per line. You get the determinant as an exact number or fraction and the full expansion.
- Nothing you type is stored
- Free
- No signup
Determinant
49
Invertible- Size
- 3 × 3
Show working
- A = [2, −3, 1; 2, 0, −1; 1, 4, 5]
- det A = a₁₁·M₁₁ − a₁₂·M₁₂ + a₁₃·M₁₃
- M₁₁ = det [0, −1; 4, 5] = 4
- M₁₂ = det [2, −1; 1, 5] = 11
- M₁₃ = det [2, 0; 1, 4] = 8
- det A = 2·4 − (−3)·11 + 1·8 = 49
How to use
- Type the matrix, one row per line (e.g. “2 -3 1”).
- Make sure every row has the same count of numbers as there are rows.
- Read the determinant and open “Show working”.
Good to know
2×2: det [a, b; c, d] = ad − bc. 3×3: expand along the first row, det A = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃, where each minor M is the 2×2 determinant left after deleting that row and column.
4×4 and larger: the matrix is reduced to upper-triangular form by row operations. The determinant is the product of the diagonal, with a sign change for every row swap.
det A = 0 means the matrix is singular: no inverse, and the system Ax = b has no unique solution. The size of |det A| is the factor by which the matrix scales area or volume.
Questions
What is the determinant of [1, 2; 3, 4]?
1×4 − 2×3 = −2.
How is a 3×3 determinant expanded?
Multiply each entry of the first row by its minor, with signs +, −, +, and add. The working shows each minor.
What does a zero determinant mean?
The rows (or columns) are dependent, so the matrix has no inverse.
Can the determinant be a fraction?
Yes, if the entries are fractions or decimals. It is shown exactly, e.g. 7/12.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.