The determinant is a scalar value computed from a square matrix that encodes important properties of the matrix.
For a 2×2 matrix [[a,b],[c,d]], the determinant is ad - bc. The determinant is zero if and only if the matrix is singular (non-invertible). The absolute value of the determinant equals the area (2D) or volume (3D) of the transformation defined by the matrix. Determinants are used to solve systems of linear equations (Cramer's rule), find eigenvalues, and determine whether a matrix is invertible.
Determinants were first studied by Japanese mathematician Seki Takakazu in 1683 and independently by Gottfried Leibniz. The term "determinant" was coined by Augustin-Louis Cauchy in 1812. Carl Friedrich Gauss used determinants in number theory. The modern theory of determinants was developed by James Joseph Sylvester and Arthur Cayley in the 19th century.