Determinants

Determinants are scalars associated with square matrices that capture how the corresponding linear transformation scales volume and, in a controlled way, preserves or reverses orientation. For an n-by-n matrix A, the determinant det(A) depends on all entries of A in a multilinear and alternating fashion and provides a compact summary of many algebraic and geometric properties of A.

The determinant of small matrices is often written in explicit formulas. For a 2-by-2 matrix [[a, b], [c, d]], det is ad − bc. For a 3-by-3 matrix [[a, b, c], [d, e, f], [g, h, i]], det is a(ei − fh) − b(di − fg) + c(dh − eg).

Key properties include: det(I) = 1 for the identity matrix; det(AB) = det(A) det(B); det(A^T) = det(A); swapping two rows (or columns) changes the sign of the determinant; multiplying a row (or column) by a scalar k multiplies det by k; adding a multiple of one row to another leaves the determinant unchanged. These rules imply that determinant is zero exactly when the matrix is singular (not invertible).

Computation methods include expansion by minors (Laplace expansion) and row reduction to triangular form, where det(A) equals the product of the diagonal entries times a sign determined by row swaps. The absolute value of the determinant equals the volume scaling factor of the linear transformation, while the sign indicates orientation.

Applications span solving linear systems (Cramer's rule requires nonzero determinant), change of variables in integration (Jacobian determinant), and characterizing invertibility and volume spanned by column vectors.