6) Let S be a symmetric, invertible matrix and let M be an invertible matrix of the same size as S. Prove the following: T 1 M¯¹ (S¯¹) "'S(SM²) ¹ = S

а 3 2 Find the value of a that make the matrix A -1 2 3 1 2 singular. a =

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.

7. For A = 310 0 1 0 find a diagonal matrix and an invertible matrix S so that 4 2 1 A=SAS. Then use these results to find