Let A and B be n times n matrices Which of the following is

Let A and B be n times n matrices. Which of the following is false? det (A) det (B) = det (AB) If lambda - 1 is a factor of the characteristic polynomial of A, then 1 is an eigenvalue of A. If A and B are row equivalent, A and B have the same eigenvalues. If A is triangular, the product of the diagonal entries of A is the determinant of A.

Solution

Answer: Third option is false (As explained below)

If A and B are row equivalent, they may or may not have same eigenvalues. One can check this by referring to any nonsingular matrix. Every such nonsingular matrix is row equivalent to an identity matrix, but matrices do not necessarily have all eigenvalues equal to 1.