If A is an invertible matrix that is orthogonally diagonalizable, show that A-1 is orthogonally diagonalizable. Identify the error(s) in the following proof: (Select all that apply:)
If A is invertible and orthogonally diagonalizable, then QTAQ = D for some invertible diagonal matrix Q and orthogonal matrix D, so that D = (QTAQ)-1 = Q-1A-1(QT)-1 = QTA-1(QT)T = QTA-1Q.
Thus, A-1 is also orthogonally diagonalizable (and by the same orthogonal matrix).
The proof should begin with the assumption that A-1 is orthogonally diagonalizable. The invertible diagonal matrix Q should be D, and the orthogonal matrix D should be Q. We cannot begin the proof with the statement that A is invertible and orthogonally diagonalizable. D = (QTAQ)-1 should be D-1 = (QTAQ)-1. D = (QTAQ)-1 should be DT = (QTAQ)-1.