Definition: In algebra, a Jacobian matrix is a special kind of square matrix that arises from a change of variable in the calculus. It's named after Carl Gustav Jacob Jacobi, who introduced it in 1854 to study the geometry and dynamics of conic sections. A Jacobian matrix is an important tool in linear algebra, especially for solving systems of differential equations or finding eigenvalues of matrices. Its properties are used extensively in various branches of mathematics and engineering, particularly in numerical analysis