Matrix Calculator
Free online matrix calculator. Perform 2×2 and 3×3 matrix addition, subtraction, multiplication, determinant, and transpose operations.
Matrices are rectangular arrays of numbers used to represent linear transformations, systems of equations, and data structures. This calculator supports addition, subtraction, multiplication, determinant, and transpose for 2×2 and 3×3 matrices — the most common sizes in school and undergraduate courses.
Key Matrix Operations
- Add/Subtract: Combine matrices element-by-element (same size required)
- Multiply: Row × Column dot products — A (m×n) × B (n×p) = C (m×p)
- Determinant: A scalar value; det ≠ 0 means the matrix is invertible
- Transpose: Flip rows and columns — (Aᵀ)ᵢⱼ = Aⱼᵢ
Determinant Formulas
For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc. For a 3×3 matrix, it is computed by cofactor expansion. A determinant of zero means the matrix is singular (no inverse), which indicates a system of equations has no unique solution. In geometry, |det| gives the area (2D) or volume (3D) scaling factor of the transformation.