Determinant Calculator — 2×2 and 3×3 Matrix
Enter the elements of a 2×2 or 3×3 matrix to calculate its determinant, see the full cofactor-expansion working, and check whether the matrix is invertible.
Matrix size
Matrix is invertible (det ≠ 0)
Matrix
Formula
Substitution
Result
- 1
Multiply main-diagonal entries
3 × 4 = 12 - 2
Multiply off-diagonal entries
2 × 1 = 2 - 3
Determinant = ad − bc
12 − 2 = 10
How does this calculator work?
For a 2×2 matrix [[a,b],[c,d]]: det = ad − bc. For a 3×3 matrix: det = a(ei−fh) − b(di−fg) + c(dh−eg). Zero determinant → singular (non-invertible). |det| = area/volume scale factor of the transformation. Enter matrix elements and get the determinant with full step-by-step working.
Formula
How this is calculated
The determinant of a square matrix is a scalar that encodes key geometric and algebraic properties of the linear transformation the matrix defines. For a 2×2 matrix [[a, b], [c, d]], the determinant is simply ad − bc — the signed area of the parallelogram spanned by the two column vectors.
For a 3×3 matrix, this calculator uses cofactor expansion along the first row: det = a × M₁₁ − b × M₁₂ + c × M₁₃, where each 2×2 minor Mᵢⱼ is the determinant of the submatrix formed by removing row 1 and the corresponding column. The alternating +/− signs follow the standard chessboard sign pattern of the cofactor matrix.
A determinant of zero means the matrix is singular: its rows (or columns) are linearly dependent, the system Ax = b has no unique solution, and no inverse exists. The absolute value of the determinant gives the area scale factor (2×2) or volume scale factor (3×3) — the unit square or cube is stretched or squished by exactly |det(A)| under the transformation.
Frequently asked questions
For matrix [[a, b], [c, d]], the determinant is ad − bc. Multiply the main-diagonal elements and subtract the product of the off-diagonal elements. For example, [[3, 2], [1, 4]] gives det = 3×4 − 2×1 = 10.
A zero determinant means the matrix is singular: rows are linearly dependent, the matrix has no inverse, and the linear system Ax = b either has no solution or infinitely many. Geometrically, the transformation collapses the space into a lower dimension.
For a 2×2 matrix, |det(A)| is the area of the parallelogram spanned by the column vectors. For a 3×3 matrix it is the volume of the parallelepiped. The sign indicates whether the orientation is preserved (positive) or reversed (negative).
Also known as
TG we-Calculate Editorial Team. (2026). Determinant Calculator — 2×2 and 3×3 Matrix [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/determinant-calculator
TG we-Calculate Editorial Team. "Determinant Calculator — 2×2 and 3×3 Matrix." TG we-Calculate. 2026. https://we-calculate.com/calculator/determinant-calculator.
TG we-Calculate Editorial Team, "Determinant Calculator — 2×2 and 3×3 Matrix," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/determinant-calculator
@misc{wecalculate_determinant_calculator, title = {Determinant Calculator — 2×2 and 3×3 Matrix}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/determinant-calculator}}, year = {2026}, note = {TG we-Calculate} }
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