Final answer:
Given that the matrix product ab is invertible, we can conclude that both matrices a and b are invertible by showing that a has a right inverse and b has a left inverse, thus, by definition, they are invertible.
Step-by-step explanation:
To show that both a and b are invertible given that the product ab is invertible, we can use the properties of matrix multiplication and inverses. Since ab is invertible, there exists a matrix (ab)^{-1} such that:
(ab)(ab)^{-1} = I
and
(ab)^{-1}(ab) = I,
where I is the identity matrix. To find the inverse of a and b, we will manipulate these equations. We can rewrite the first equation as:
a(b(b^{-1}a^{-1})) = I,
which implies that b^{-1}a^{-1} is the right inverse of a. Similarly, from the second equation, we can rewrite it as:
((a^{-1}b^{-1})b)a = I,
which implies that a^{-1}b^{-1} is the left inverse of b. By definition, if a matrix has both a left inverse and a right inverse, then the matrix is invertible, and the left and right inverses are equal. Hence, a has a right inverse and b has a left inverse, therefore both a and b must be invertible.