Matrix Calculator – Determinant, Inverse & More

The identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. For a 2×2 identity: [[1,0],[0,1]]. Multiplying any compatible matrix A by the identity matrix gives back A unchanged — making it the matrix equivalent of multiplying a number by 1. It is the neutral element of matrix multiplication.

Yes — you can multiply a 3×2 matrix by a 2×4 matrix because the inner dimensions both equal 2. The result is a 3×4 matrix (the outer dimensions). The general rule: an m×n matrix multiplied by an n×p matrix produces an m×p result. If the inner dimensions do not match, the multiplication is undefined and has no meaning.

A singular matrix has a determinant of 0 and has no inverse. Geometrically, a singular transformation "flattens" space — reducing a 2D plane to a line, or a 3D space to a plane. Singular matrices arise in systems of equations with no unique solution (either no solutions or infinitely many).

The transpose of a matrix A (written Aᵀ) is obtained by flipping rows and columns. If A = [[1,2,3],[4,5,6]], then Aᵀ = [[1,4],[2,5],[3,6]]. An m×n matrix becomes an n×m matrix when transposed.