does inverse commute with transpose

Matrix Question: Does inverse commute with transpose? PhL 11.26.09 Here we are, once again, doing 4th grade math and being confused, but what else is new? 1. What is the minor? minor(Mab) = determinant of (matrix obtained by crossing out row a and row b) = a number Let's define a matrix called minor(M) which has these matrix elements [minor(M)]ab = minor(Mab) = a number Then it follows that [minor(MT)]ab = minor( (MT)ab) = minor (Mba) = [minor(M)]ba = ([minor(M)]T)ab Therefore we have shown that minor(MT) = [minor(M)]T // a matrix equation "The transpose of the minor-matrix of M (RHS) is the minor-matrix of the transpose of M (LHS)." or "The minor operation commutes with the transpose operation". 2. What is the cofactor? cof(Mab) = (-1)a+b minor(Mab) = a number Let's define a matrix called cof(M) which has these matrix elements [cof(M)]ab = cof(Mab) =(-1)a+b [minor(M)]ab Then if follows that [cof(MT)]ab = cof((MT)ab) =(-1)a+b [minor(MT)]ab = (-1)a+b ([minor(M)]T)ab = (-1)a+b [minor(M)]ba = (-1)b+a [minor(M)]ba = [cof(M)]ba = ([cof(M)]T)ab we have then shown that cof(MT) = [cof(M)]T // a matrix equation " Cofactor operation commutes with T operation " 3. What is the inverse? M-1 = cof(M)/det(M) Then we have: (MT)-1 = cof(MT)/det(MT) = cof(MT)/det(M) = [cof(M)]T / det(M) But on the other hand we have (M-1)T = [cof(M)]T/det(M) Thus we have proved that (MT)-1 = (M-1)T which says that "transpose commutes with inverse".