A characteristic polynomial is a mathematical expression that helps determine the properties of a square matrix A. It is derived from the determinant of the matrix subtracted by a variable λ times the identity matrix I. The resulting polynomial, typically denoted as p(λ), provides important information about the matrix, such as its eigenvalues, which are the values of λ that make the polynomial equal to zero. The roots of the characteristic polynomial correspond to the eigenvalues of the matrix A. These eigenvalues are crucial in various applications, including stability analysis in differential equations and quantum mechanics. By studying the characteristic polynomial, mathematicians and scientists can gain insights into the behavior of linear transformations represented by the matrix.