Question

Is the product of two invertible matrices also invertible?

Answer 1

Yes

Explanation:

Matrix multiplication is associative, so `(AB)C = A(BC)` and we can just write `ABC` unambiguously.

Suppose `A` and `B` are invertible, with inverses `A^-1` and `B^-1`. Then `B^-1A^-1` is the inverse of `AB`:

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