Matrix Determinant Calculator
Determinant of a 2×2 or 3×3 matrix.
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Formula
det [[a,b],[c,d]] = ad − bc · 3×3: cofactor expansion along row 1
The determinant measures how a linear transformation scales signed area (2D) or volume (3D). Zero determinant means the matrix squashes space to a lower dimension — it's singular and not invertible.
Worked example
For [[1, 2], [3, 4]], det = 1×4 − 2×3 = −2. The negative sign means orientation flips under this transformation.
Where this can give the wrong answer
- Switch to 3×3 mode for larger matrices — entries a₁₃, a₂₃, a₃₁, a₃₂, a₃₃ matter only in that mode.
- Integer-looking inputs with decimals (1.0000001) can produce tiny floating-point noise in the determinant.
- Determinant zero doesn't display 'undefined' — it correctly returns 0, signalling a singular matrix.
FAQ
- Cramer's rule for solving systems, checking invertibility, computing eigenvalues in theory, and Jacobian factors in calculus. In practice, det = 0 tells you the system has no unique solution.