For invertible matrices A and B of the same order, (AB)−1=B−1A−1
The inverse operation reverses the order of the product.
To verify this is the correct inverse, multiply (AB) with (B−1A−1):
(AB)×(B−1A−1)
=A(BB−1)A−1
=A(I)A−1
=AA−1
=I
Since the product equals the identity matrix I, (AB)−1=B−1A−1
This result extends to products of multiple matrices:
(ABC)−1=C−1B−1A−1
The order is always reversed when taking the inverse of a product.
Therefore, (AB)−1=B−1A−1