Exercise (i) Show that, for any AFmn, one always has rank(MAN)=rank(A) whenever M,N are nonsingular square matrices of suitable orders such that the product MAN is well-defined. (ii) Let A be a nonsquare matrix. Show that the matrix equation AX=I cannot have solution X if A has more rows than columns, and the matrix equation YA=I cannot have solution Y if A has more columns than rows. Here I is an identity matrix of some suitable order. (iii) Suppose A,B are square matrices of the same order and satisfy AB=I. Show that BA=I. (Note: this result implies that we need to check only one of the two equations AB=I,BA=I when we want to verify that two square matrices A,B are inverses of each other.)