Simplifying matrices

Question

Given that A and B are non-singular square matrices, simplify:

 

AB(A^-1B)^-1

 

(what does the question mean by square matrices?) 

Answer 1

A matrix is "square" when the number of rows is the same as the number of columns.  If a matrix is square, then it may have an inverse.  If the matrix has an inverse, the matrix is said to be "nonsingular".

 

Assuming that A and B are both nxn nonsingular matrices,  

 

AB(A^-1B)^-1 = ABB^-1(A^-1)^-1

 

                   = AIA, where I is the nxn identity matrix

 

                   = A²