Let A be n n matrix True or false AT has the same eigenvect

Let A be n × n matrix. True or false:

A^T has the same eigenvectors as A.

Solution

We have det(A^T- I) = det(A^T- I^T) = det(A-I)^T = det(A -I)

solutions of det(A –I) = 0 is also solution of det(A –I)T = 0 and vice versa.

Hence A and A^T have the same eigenvalues.

True