Matrix Calculator

Matrices are rectangular arrays of numbers used extensively in linear algebra, physics, computer graphics, machine learning, and engineering. The three fundamental operations covered by this calculator are:

Determinant: A scalar value computed from a square matrix that encodes important properties. For a 2x2 matrix [[a,b],[c,d]], det = ad - bc. For a 3x3 matrix, the determinant is computed using cofactor expansion along the first row. A zero determinant means the matrix is singular and has no inverse.

Inverse: The inverse of matrix A is a matrix A^(-1) such that A x A^(-1) = I (identity). For a 2x2 matrix, A^(-1) = (1/det) x [[d,-b],[-c,a]]. For a 3x3 matrix, the inverse is computed using the adjugate (transpose of the cofactor matrix) divided by the determinant. Only matrices with non-zero determinants are invertible.

Transpose: The transpose swaps rows and columns: element at position (i,j) moves to (j,i). The transpose is always defined and does not require a non-zero determinant.