Determinant Calculator

How to Use This Calculator

1. Select matrix size and enter values — determinant and steps appear instantly.
2. Use 3x3 or 4x4 tabs for larger matrices.
3. Use Professional for Cramer's rule and multiplicative properties.

Formula

Example

Frequently Asked Questions

• How do you calculate a 2x2 determinant? ▼
  For a 2×2 matrix [[a, b], [c, d]], the determinant is det(A) = ad − bc — multiply the main diagonal and subtract the product of the anti-diagonal. Example: A = [[1, 2], [3, 4]]. det(A) = (1)(4) − (2)(3) = 4 − 6 = −2. Another example: A = [[5, 3], [2, 4]]. det(A) = (5)(4) − (3)(2) = 20 − 6 = 14. A common mistake is subtracting in the wrong order — some students compute bc − ad instead of ad − bc. Also note that swapping two rows of a matrix negates the determinant, so [[3, 4], [1, 2]] gives det = 6 − 4 = 2, which is −(−2) = 2 as expected.
• How do you calculate a 3x3 determinant? ▼
  Use cofactor expansion along the first row: det(A) = a11·(a22·a33 − a23·a32) − a12·(a21·a33 − a23·a31) + a13·(a21·a32 − a22·a31). The pattern is + − + for the cofactor signs. Example: A = [[1,2,3],[4,5,6],[7,8,9]]. det = 1·(5·9−6·8) − 2·(4·9−6·7) + 3·(4·8−5·7) = 1·(45−48) − 2·(36−42) + 3·(32−35) = 1·(−3) − 2·(−6) + 3·(−3) = −3 + 12 − 9 = 0. A common pitfall is sign errors: the middle term always gets a minus sign regardless of the actual element values. You can expand along any row or column for efficiency — choose the one with the most zeros.
• What does det=0 mean? ▼
  If det(A) = 0, the matrix is called singular or non-invertible. Geometrically, this means the row vectors are linearly dependent — they lie in a lower-dimensional space, collapsing the parallelogram/parallelepiped to zero area/volume. Algebraically, the system Ax = b has either no solution or infinitely many solutions (never exactly one). For a 2×2 matrix, det = 0 means one row is a scalar multiple of the other. Example: A = [[2, 4], [1, 2]] has det = 2·2 − 4·1 = 0, and indeed row 2 = (1/2) × row 1. A common pitfall is assuming a small determinant means a matrix is nearly singular — but numerical near-singularity (ill-conditioning) is measured by the condition number, not the determinant magnitude alone.
• What is the geometric meaning? ▼
  The absolute value of the determinant gives the scale factor by which a matrix transformation changes area or volume. For a 2×2 matrix, |det(A)| equals the area of the parallelogram spanned by the two row (or column) vectors. For a 3×3 matrix, |det(A)| equals the volume of the parallelepiped. Example: A = [[3, 0], [0, 2]] has det = 6, meaning it stretches areas by a factor of 6. The sign of the determinant indicates orientation: a positive determinant preserves orientation; a negative determinant reverses it (a reflection or odd number of reflections). This geometric interpretation is foundational in multivariable calculus for change-of-variables formulas (the Jacobian determinant).
• What is the multiplicative property? ▼
  The determinant satisfies det(AB) = det(A) × det(B) for square matrices of the same size. This means multiplying matrices corresponds to multiplying their determinants — a useful property when checking if a product is invertible. Additional key properties: det(Aᵀ) = det(A) (transpose preserves determinant); det(A⁻¹) = 1/det(A) (inverse has reciprocal determinant); det(cA) = cⁿ·det(A) for an n×n matrix (scalar multiple). Example: If det(A) = 3 and det(B) = −2, then det(AB) = −6. A common pitfall is thinking det(A + B) = det(A) + det(B) — that is false. Determinants are multiplicative but NOT additive.

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