Matrix Determinant Calculator

The determinant is a scalar value computed from the elements of a square matrix. It has many important properties: it tells you whether a matrix is invertible (det ≠ 0 means invertible), it represents the scaling factor of the linear transformation the matrix represents, and the absolute value represents the volume scaling in geometric transformations.

For a 2x2 matrix [[a, b], [c, d]], the determinant is ad - bc. This is a simple formula: multiply the diagonal elements and subtract the product of the off-diagonal elements.

For a 3x3 matrix, there are several methods. The Sarrus rule is one approach. The cofactor expansion method expands along a row or column, reducing it to three 2x2 determinants. This calculator uses cofactor expansion along the first row.

If the determinant is zero, the matrix is singular, meaning it is not invertible. Geometrically, it means the matrix collapses the space into a lower dimension. The rows and columns are linearly dependent.

Determinants are used to check if a matrix is invertible, to solve systems of linear equations (Cramer's rule), to compute volumes and areas in geometry, to find eigenvalues, and in computer graphics and physics simulations.