Final answer:
To prove ab is invertible with (ab)^{-1} = b^{-1}a^{-1}, it is shown that the product of ab and its proposed inverse results in the identity matrix, using associative properties of matrix multiplication and the concept of matrix invertibility.
Step-by-step explanation:
To prove that the product of two invertible matrices ab is also invertible with (ab)^{-1} = b^{-1}a^{-1}, we need to demonstrate that when ab is multiplied by b^{-1}a^{-1}, the result is the identity matrix. We can start by using the associative property of matrix multiplication to group a and b separately with their inverses:
(ab)(b^{-1}a^{-1}) = a(bb^{-1})a^{-1}
Since b is invertible, bb^{-1} is the identity matrix I. This simplifies our expression to:
aIa^{-1} = aa^{-1} = I
Here, aa^{-1} is also the identity matrix, confirming that b^{-1}a^{-1} is indeed the inverse of ab. Therefore, ab is invertible and its inverse (ab)^{-1} is equal to b^{-1}a^{-1}, as required.