For any two elements A and B that form AB, neither A nor B have to be

[Gear300]

For any two elements A and B that form AB, neither A nor B have to be invertible for (AB)^-1 to exist, right?

 

[defunc]

Both have to be invertible. Inv(AB) = inv(B)inv(A).

 

[Mark44]

Right. For example..
\left[ <br /> \begin{array}{c c c}<br /> 1&amp;0&amp;0\\<br /> 0&amp;1&amp;0<br /> \end{array}<br /> \right]<br /> \left[ <br /> \begin{array}{c c}<br /> 1&amp;0\\<br /> 0&amp;1\\<br /> 0&amp;0<br /> \end{array}<br /> \right]~=~\left[ <br /> \begin{array}{c c}<br /> 1&amp;0\\<br /> 0&amp;1\\<br /> \end{array}<br /> \right]<br />

The matrix on the right is clearly invertible, while the two matrices in the product aren't event square, let alone invertible.

 

[Mark44]

Both have to be invertible. Inv(AB) = inv(B)inv(A).

See my counterexample.

 

[Gear300]

Thanks for the reply. So that would mean that Inv(AB) = inv(B)inv(A) iff inv(B) and inv(A) exist (whereas Inv(AB) may exist without a defined inv(A) or inv(B)), right?

 

[pbandjay]

Looks correct to me. The condition that I am familiar with is:

If A and B are invertible matrices of the same size, then AB is invertible and (AB)^-1 = B^-1A^-1.

This is easy to prove by showing that (AB)(B^-1A^-1) = A(BB^-1)A^-1 = AIA^-1 = AA^-1 = I.

As far as the other way of your "iff", if (AB)^-1 = B^-1A^-1 is given then it seems to me that the existence of B^-1 and A^-1 would directly follow since they are used in the initial condition.

 

[Gear300]

I see. Thanks again for the replies. I have an additional question:

For a matrix C that is not a square matrix, there is no defined inverse; however, it is possible that there is a left inverse A and a right inverse B, in which A =/= B, for the matrix C, right?

 

[Mark44]

Yes. See the wikipedia article here, under the section titled Matrices.

 

[Gear300]

Thanks for the link.

So in the case of linear systems Ax = b, I suppose it wouldn't always be possible to use the left inverse of A to isolate x as a general method since the left inverse of A does not necessarily exist. How would one isolate x in these linear systems (other than parametrization of x)?