Determinant Calculator — Step-by-Step Solutions

The determinant is a single number computed from a square matrix. Geometrically, it is the factor by which the matrix scales area (2x2) or volume (3x3), and its sign tells you whether orientation is flipped. If the determinant is zero, the matrix squashes space into a lower dimension and cannot be undone.

For a 2x2 matrix [[a, b], [c, d]] the determinant is ad - bc. For larger matrices the calculator uses cofactor expansion along the first row: multiply each entry of the row by its cofactor and add, alternating signs +, -, +, -. Each cofactor is the determinant of the smaller matrix left when that entry row and column are removed, so a 3x3 reduces to three 2x2 determinants and a 4x4 to four 3x3 determinants.

Useful properties: swapping two rows flips the sign; multiplying a row by k multiplies the determinant by k; adding a multiple of one row to another changes nothing; det(AB) = det(A) x det(B); and a triangular matrix has a determinant equal to the product of its diagonal entries.

A matrix is invertible exactly when its determinant is not zero. The determinant also appears in Cramer rule for solving linear systems and in the characteristic polynomial used to find eigenvalues.