What Is a Determinant?

Matrices are mathematical objects that transform shapes. The determinant of a square matrix A, denoted |A|, is a number that summarizes the effect A has on a figure’s size and orientation. If [a b] is the top row vector for A and [c d] is its bottom row vector, then |A| = ad-bc.

A determinant encodes useful information about how a matrix transforms regions. The absolute value of the determinant indicates the scale factor of the matrix, how much it stretches or shrinks a figure. Its sign describes whether the matrix flips figures over, yielding a mirror image. Matrices can also skew regions and rotate them, but this information is not provided by the determinant.

Arithmetically, the transforming action of a matrix is determined by matrix multiplication. If A is a 2 × 2 matrix with top row [a b] and bottom row [c d], then [1 0] * A = [a b] and [0 1] * A = [c d]. This means that A takes the point (1,0) to the point (a,b) and the point (0,1) to the point (c,d). All matrices leave the origin unmoved, so one sees that A transforms the triangle with endpoints at (0,0), (0,1), and (1,0) to another triangle with endpoints at (0,0), (a,b), and (c,d). The ratio of this new triangle’s area to the original triangle’s is equal to |ad-bc|, the absolute value of |A|.

The sign of a matrix’s determinant describes whether the matrix flips a shape over. Considering the triangle with endpoints at (0,0), (0,1), and (1,0), if a matrix A keeps the point (0,1) stationary while taking the point (1,0) to the point (-1,0), then it has flipped the triangle over the line x = 0. Since A has flipped the figure over, |A| will be negative. The matrix does not change the size of a region, so |A| must be -1 to be consistent with the rule that the absolute value of |A| describes how much A stretches a figure.

Matrix arithmetic follows the associative law, meaning that (v*A)*B = v*(A*B). Geometrically, this means that combined action of first transforming a shape with matrix A and then transforming the shape with matrix B is equivalent to transforming the original shape with the product (A*B). One can deduce from this observation that |A|*|B| = |A*B|.

The equation |A| * |B| = |A*B| has an important consequence when |A| = 0. In that case the action of A cannot be undone by some other matrix B. This can be deduced by noting that if A and B were inverses, then (A*B) neither stretches nor flips any region, so |A*B| = 1. Since |A| * |B| = |A*B|, this last observation leads to the impossible equation 0 * |B| = 1.

The converse claim can also be shown: if A is a square matrix with nonzero determinant, then A has an inverse. Geometrically, this is the action of any matrix that does not flatten a region. For example, squishing a square into a line segment can be undone by some other matrix, called its inverse. Such an inverse is the matrix analog of a reciprocal.