True or Falso and explain why If A and B are Invertible n x

True or Falso and explain why

If A and B are Invertible n x n matrices, then so is AB

Solution

true

the inverse of AB is B^-1A^-1

we know A and B are invertible nxn matrix

then there exist A^-1 and B^-1 nxn matrix

now B^-1A^-1 also exist and is a nxn matrix

we know the definition of inverse XY=I(identity matrix) then Y is inverse of X

hence (AB)(B^-1A^-1) = A (BB^-1) A^-1 = A (I) A^-1 = AA^-1=I

hence AB is invertible with inversion matrix = B^-1A^-1