determinantA

The determinant is a scalar attribute of a square matrix that encodes various properties of the linear transformation it represents. It can be interpreted as the factor by which the transformation scales volumes and as an indicator of whether the transformation preserves orientation.

For an n by n matrix A, the determinant can be defined by the Leibniz formula det(A) = sum over all permutations σ of S_n of sgn(σ) product_i a_{i, σ(i)}. In practice, direct expansion is used only for small matrices. More efficient methods rely on LU or QR decomposition: after performing row operations to obtain an upper triangular form, det(A) equals the product of the diagonal entries multiplied by −1 raised to the number of row interchanges.

Several key properties accompany the determinant. It is unchanged by transposition det(A^T) = det(A); it is multiplicative det(AB) = det(A) det(B); det(kA) = k^n det(A) for a scalar k; det(I) = 1 and det(0) = 0. A matrix is invertible exactly when det(A) ≠ 0, and det(A^{-1}) = 1/det(A). Geometrically, the absolute value of det(A) gives the volume scaling factor of the linear transformation, and the sign indicates the orientation of the transformed space.

The determinant also relates to eigenvalues: det(A) equals the product of A’s eigenvalues (counting multiplicities). It plays a central role in solving linear systems, changes of variables in integration, and in characterizing matrix behavior.