If is the characteristic polynomial of an matrix A, and , then A is not invertible.

Question

If  is the characteristic polynomial of an  matrix A, and  , then A is not invertible.

Quizplus · Accepted Answer

The characteristic polynomial is a polynomial in λ that is obtained from the matrix A by replacing λI - A with λ, where I is the identity matrix of the same size as A. The roots of this polynomial are the eigenvalues of A. If any root is zero, then A is not invertible. In this case, the polynomial has a factor of λ, which means that one of the eigenvalues is zero, and hence A is not invertible. Therefore, the answer is A.

If Is the Characteristic Polynomial of an Matrix