1. Let A(t) be an anti-symmetric matrix for all t. Let M(t) be a matrix-valued function satisfying M' (t) = M(t)A(t). Show that M (t) M (t)T is a constant matrix-valued function. 2. Recall that given a matrix Ae Matn,n (R), the exponential defined by o 1 k! eA = is a well-defined element of Matn,n (R). (1) Prove that (et) = AeAt. (2) Is is true that d (eB(t)) = B'(t)eB(t) dt for all matrix-valued functions B: R Matn,n(R)? k=0 3. Identify Mat2,2 (R) = R and hence use the standard dot product on Mat2,2 (R). (1) Let A be an invertible matrix, A GL(2, R) Matn,n (R). What is the angle between A and A-? (2) Are A and A- ever perpendicular? (3) Are they ever of equal norm? and 4. Using the chain rule, compute the derivative of the maps Matn,n (R) h :Matn,n(R) A A-3 k:Matn,n (R) A An Hint: Write the functions as compositions of simpler functions. Matn, n (R)