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.