What is the characteristic polynomial?

The characteristic polynomial is defined as a polynomial whose roots correspond to the eigenvalues of a matrix. This polynomial is derived from the determinant of the matrix subtracted by a scalar multiple of the identity matrix, expressed mathematically as ( p(\lambda) = \det(A - \lambda I) ), where ( A ) is the given matrix, ( \lambda ) represents the eigenvalues, and ( I ) is the identity matrix.

When you find the roots of the characteristic polynomial, you determine the values of ( \lambda ) for which the determinant is zero, which indicates the eigenvalues of the matrix. Understanding the relationship between the characteristic polynomial and eigenvalues is essential in linear algebra, as it helps in many applications such as stability analysis, systems of differential equations, and more.

The other options do not correctly capture the significance of the characteristic polynomial. For instance, it does not represent matrix dimensions, nor is it related to matrix addition or eigenvectors. Instead, it directly concerns the scalar quantities (eigenvalues) that provide insights into the properties and behavior of the matrix.