3.1.P15 Let n ≥ 2, let x, y € C" be given nonzero vectors, and let A = xy*. (a) Show if y*x #0 and 0 0 that the Jordan canonical form of A is B On-2, in which B = B = J₂(0) if y*x = 0. (b) Explain why a rank-one matrix is diagonalizable if and only if its trace is nonzero.

O. product there exists positave depnite matrix be a scalav RÉ Prove that a Eymmetric, . let = a X,Y,+ b (x, yet Xa% +C X. Ye. =>o Comment meaus

2. (i) Let A be an invertible matrix. Assume that A = A-1, What are the possible values for det(A)? (ii) Let B be a skew-symmetric matrix over R, i.e. AT = -B and of order 2n-1. Show that det(B) = 0. (iii) Show that if a square matrix C is hermitian, i.e. C* = C then det(C) is a %3D real number.

1. Let matrix A, B, A + B, A + B are nxn non-singular matrices В, matrix means it has an inverse), prove: B(A + B) + (A' +B)' B = I