How matrix multiplication works
Each entry of A × B is the dot product of a row of A with a column of B: multiply matching entries and add them up.
[1 2] [5 6] [1·5+2·7 1·6+2·8] [19 22]
[3 4] × [7 8] = [3·5+4·7 3·6+4·8] = [43 50]
[3 4] × [7 8] = [3·5+4·7 3·6+4·8] = [43 50]
Determinant and inverse of a 2 × 2 matrix
A = [a b; c d]
det(A) = ad − bc
A⁻¹ = (1 ÷ det) × [d −b; −c a]
det(A) = ad − bc
A⁻¹ = (1 ÷ det) × [d −b; −c a]
For larger matrices the calculator uses row reduction (Gaussian elimination with partial pivoting), which is fast and numerically stable.
Where matrices are used
- Systems of equations: solve several equations at once. The RREF calculator shows each row step.
- Computer graphics: every rotation, scale and 3D projection is a matrix multiplication.
- Data and AI: spreadsheets of numbers, images and neural networks are all stored and processed as matrices.
Last reviewed October 2026.