math qestions:

1. Let K be a field, A E Km\" and B E F\". [a] Prove that {AB}T = BTAT. [Hint: Using the remark on page 7'9 of the lecture notes, one can write down formulae STAT ] for c5113 and 1':- (pl Show that column rank AB 3 column rankA row rank AB 1: row rank B and deduce that rank AB 1: min[rank 31,1111]: B}. [Hint: Read the remark on page T9 of the lecture notes] 2. Determine for which values of A E R the real matrix ll A1 cpln3.- 1 A i] i] is invertible. and compute the inverse matrix Ail for these values of A. [Hint: Convert A; into echelon form using elementary row operations. The matrix A; is invertible if this echelon form has no rows which are identically zero. In this case A), can be converted into the identityr matrix L; using elementary row operations. The matrix Ail is obtained by applying these elementary row operations to L, in the same order_]