Question 7 [15 marks in total] 1. [5 marks] (Challenging) According to the spectral theorem, every n xn real symmet- ric matrix can be diagonalized by an orthogonal matrix. Does the converse hold? In other words, suppose that A = ODO-1for some (real) orthogonal matrix O and (real) diagonal matrix D, do we know that A is symmetric? 2. [10 marks] (Challenging) A real n xn matrix A is called skew-symmetric if A" = -A. Daniel convinced himself that a real skew-symmetric matrix cannot be invertible with the following "proof". Proof. Let A be an n x n real skew-symmetric matrix. Let A be an eigenvalue of A and v be an eigenvector of A associated with A. Then UT Av = U(Av) = UP(Av) = Alu|2; DAU = (ATv)Tv = (-Av) v = (-X)v = -Alu|2. The second equality in the second line uses the assumption that A is skew-symmetric. Comparing the two lines reveals that Av|? = -Av|2. Since v *0 by the definition of eigenvectors, A = 0. Now Av = Av = 0, so the non-zero vector v belongs to the kernel of A, showing that A is not invertible. 0 Determine whether the "proof" is correct. If the "proof" is incorrect, identify all the errors and determine whether the proposition that no real skew-symmetric ma- trix is invertible is true. (Hint: it may be useful to write down some concrete skew-symmetric matrices and examine whether the proposition and each step of its "proof" works for them.)