Add, multiply, transpose and invert matrices, and find determinants.
A matrix is a rectangular array of numbers. The determinant of a square matrix is a scalar that encodes many properties: a matrix is invertible if and only if its determinant is non-zero. The inverse A⁻¹ satisfies A × A⁻¹ = I (the identity matrix). Transposing swaps rows and columns. Matrix multiplication is not commutative: A × B ≠ B × A in general.
2×2 determinant
det [[a,b],[c,d]] = ad − bc
2×2 inverse
[[a,b],[c,d]]⁻¹ = (1/(ad−bc)) × [[d,−b],[−c,a]]
Matrix multiplication element
(AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ
det = (1)(4) − (2)(3) = 4 − 6 = −2. Since it's non-zero, the matrix is invertible.
A singular matrix has determinant zero and cannot be inverted. It means the rows (or columns) are linearly dependent — one row can be expressed as a combination of the others. Geometrically it means the transformation collapses space onto a lower dimension.
No. Matrix multiplication A × B requires the number of columns in A to equal the number of rows in B. A 2×2 times a 3×3 has dimension mismatch. This calculator handles square 2×2 and 3×3 matrices only.