Inverses

**Gear300** · Nov4-09, 06:49 PM

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** · Nov4-09, 06:58 PM

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

**Mark44** · Nov4-09, 07:01 PM

Right. For example..
$LaTeX Code: \\left[ \\begin{array}{c c c} 1&0&0\\\\ 0&1&0 \\end{array} \\right] \\left[ \\begin{array}{c c} 1&0\\\\ 0&1\\\\ 0&0 \\end{array} \\right]~=~\\left[ \\begin{array}{c c} 1&0\\\\ 0&1\\\\ \\end{array} \\right] $

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

**Mark44** · Nov4-09, 07:04 PM

Originally Posted by defunc

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

See my counterexample.

**Gear300** · Nov4-09, 07:14 PM

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** · Nov4-09, 07:43 PM

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** · Nov5-09, 12:26 PM

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** · Nov5-09, 01:06 PM

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

**Gear300** · Nov5-09, 01:23 PM

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)?