The determinant

The determinant det(A) maps a square matrix to a real number. It equals the product of all the eigenvalues of A (the trace is their sum).

Its absolute value measures how much multiplying by A expands or contracts space:

• det(A) = 0: space is squashed completely along at least one dimension and loses all its volume. The matrix is singular.
• det(A) = 1: the transformation preserves volume.