True or false:

Suppose A E Mnxn (C) is a matrix whose rows form an orthonormal basis for Cn with respect to the standard inner product. Then A must be a unitary matrix.Is it possible to have a symmetric matrix A E Mnxn(R) such that where is the identity matrix? (Hint: think about eigenvalues.) Consider the following attempted proof that every orthogonally diagonalizable matrix A E Mnxn (C) is symmetric. Suppose A E Mnxn (C) is orthogonally diagonalizable. By definition, this means there is some orthogonal matrix P and diagonal matrix D such that D = P- AP. Re-writing this, we have PDP- = A, and since P is orthogonal, we havep-1 = pT, so that PDP = A. Taking transposes, we have A' = (PDPT)T = (PT)TDT PT = PDT PT = PDP = A, so that A is symmetric. What, if anything, is wrong with this proof?O The proof is correct as given. 0 The definition of orthogonal diagonalization is given incorrectly. O The expression PDPT = A is derived incorrectly. O For a diagonal matrix with complex entries, the equality DT = D does not hold. Suppose we know A E M3X3 ( R) is an orthogonally diagonalizable matrix, and we are given that V1= and V2 = are eigenvectors of A. Given this, a third linearly independent eigenvector of A must exist in the formmust exist in the form for some integers and What are the values of and b