6. The following is a quiz in a linear algebra course. Prove or disprove the following statements. (a) 0)) If a square matrix A has real entries, then its eigenvalues are real. If A is a 2 X 2 matrix, then A must be similar to AT. The solutions given by a student are shown below: (a) We will prove the statement as follows. Let A be an eigenvalue of A and x be a corresponding eigenvector. Note that Z = A since A has real entries. We have Ax=Ax=Ex=(A_rc)=EAxT)=Ix. Since 3: is an eigenvector, it must be non-zero. Hence Ax = Ex implies in = A, so A is real. We will disprove the statement as follows. Suppose the statement is true, is. A is similar to AT for every 2 X 2 matrix A. Then there exists an invertible matrix P such that A = P'lATP, i.e. PA = ATP. Let A = [Z :3] and P = E i]. Then PA 2 ATP for all A implies [1516+Cf be+df]_ ee+cg ef+ch] og+ch by-i-dh _ be+dg bf-l-dh for any choice of a, b, c and d. Comparing the (LU-entry we get of : cg for any choice of :3, so setting c =-- 1 gives 3' = 9. Comparing the (1, Ej-entry we get be+ [if : efi-ch for any choice of a, b, c and d, so setting (a,b,c,d) = (1,1,D,U) gives 6 = f while setting (a,b,c,d) m (0, 1, 1,0) gives 6 == h. It follows that e = f m g = h, contradicting P being invertible. It follows that the original statement is false