A is an n times n matrix Check the true statements below A n

A is an n times n matrix. Check the true statements below: A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a nontrivial solution x. To find the eigenvalues of A, reduce A to echelon form. A matrix A is not invertible if and only if 0 is an eigenvalue of A. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy If Ax = lambda x for some vector x, then lambda is an eigenvalue of A.

Solution

(A) True, because the solution of the equation (A-cI)x = 0 is nontrivial.
(B) False. This will change the matrix and hence the eigenvalues and the eigenvectors.
(C) True, because |A| = 0 if and only if 0 is an eigenvalue.
(D) True. To check whether a given non-zero vector x is an eigenvector, you should
find the matrix-vector product Ax and check whether Ax = cx for some scalar c.
(E) False. True if x is a non-zero vector.