Question

Why must the product of two invertible matrices be invertible?

Answer 1

If `A` has inverse `A^(-1)` and `B` has inverse `B^(-1)`, then `AB` has inverse `B^(-1)A^(-1)`

Explanation:

`(AB)(B^(-1)A^(-1)) = A(BB^(-1))A^(-1) = AIA^(-1) = A A^(-1) = I`

`(B^(-1)A^(-1))(AB) = B^(-1)(A^(-1)A)B = B^(-1)IB = B^(-1)B = I`