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Determinant calculator

Type a square matrix, one row per line with the numbers separated by spaces, and get the determinant. Fractions such as 1/2 are allowed and the answer is exact.

One row per line, numbers separated by spaces. Fractions such as 1/2 are allowed.

Fill in the fields; the answer appears here straight away.

How it works

  1. Enter the values

    Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.

  2. Instant answer

    The answer appears as you type, with the most important intermediate values.

  3. See the working

    Under "How it's worked out" you see the steps, handy for checking your own calculation.

How it is calculated

For a 2×2 matrix the determinant is ad − bc. Larger matrices are reduced to a triangular form with row operations (Gaussian elimination); the determinant is then the product of the pivots, with a minus sign for every row swap. That is much faster than expanding by cofactors and works up to 10×10.

What the determinant tells you

  • det = 0: the rows are dependent, the matrix has no inverse and a system of equations with this matrix has no unique solution.
  • det ≠ 0: the matrix is invertible. The absolute value is the factor by which the matrix scales areas (2×2) or volumes (3×3).

To find the inverse itself, use the inverse matrix calculator.

Worked examples

  • 2×2: for [[3, 8], [4, 6]] the determinant is 3 × 6 − 8 × 4 = −14.
  • 3×3: for the default [[2, −1, 0], [1, 3, 2], [0, 1, 4]], expand along the first row: 2 × (3 × 4 − 2 × 1) − (−1) × (1 × 4 − 2 × 0) + 0 = 20 + 4 = 24.
  • Triangular: [[2, 7, 1], [0, 3, 5], [0, 0, 4]] has determinant 2 × 3 × 4 = 24.

Area and volume

The vectors (3, 1) and (1, 2) span a parallelogram of area |3 × 2 − 1 × 1| = 5. The same determinant scales areas in the plane under a transformation, and a 3×3 determinant gives the volume of the parallelepiped spanned by three vectors.

Useful rules

Swapping two rows flips the sign. Multiplying one row by k multiplies the determinant by k. Adding a multiple of one row to another changes nothing. And det(A · B) = det(A) · det(B). For solving a system of equations, a non-zero determinant tells you there is exactly one solution; use the system of equations solver to find it, or the inverse matrix calculator.

Frequently asked questions

What does a determinant of 0 mean?

The rows are linearly dependent: one is a combination of the others. The matrix has no inverse, and a system of equations with this matrix has no unique solution.

Can a determinant be negative?

Yes. A negative determinant means the transformation flips orientation, like a mirror image. The absolute value is still the scaling factor of areas or volumes.

How large can the matrix be?

Square matrices up to 10×10. Non-square matrices have no determinant, and the calculator tells you so.

What is the determinant of a triangular matrix?

The product of the diagonal entries. That is also what the elimination method ends with, after adjusting the sign for any row swaps.