Thread: invertible matrix

I was wondering if its possible to know if a matrix is invertible without row reducing it in attempt to make it I. If the matrix is not nxn, it is never invertible?

Originally Posted by Kuma

I was wondering if its possible to know if a matrix is invertible without row reducing it in attempt to make it I. If the matrix is not nxn, it is never invertible?

It doesn't make sense to speak of invertibility for non matrices. In particular, a matrix which is can be thought of a linear transformation and so if is invertible, then the associated map is an isomorphism, which implies that .

Originally Posted by Drexel28

It doesn't make sense to speak of invertibility for non matrices. In particular, a matrix which is can be thought of a linear transformation and so if is invertible, then the associated map is an isomorphism, which implies that .

great explanation, thanks. Now in an nxn case, can you simply tell if a matrix is invertible just by looking at it without trying to use the gauss jordan method to find its inverse?

Originally Posted by Kuma

great explanation, thanks. Now in an nxn case, can you simply tell if a matrix is invertible just by looking at it without trying to use the gauss jordan method to find its inverse?

Well, since you liked the last explanation, perhaps one in the same vein. Pretend for a second that a matrix really is just a linear transformation , then one only has to check any of the following equivalent conditions:


1) is injective


2) is surjective


3) does not have zero as an eigenvalue


4)


And there are many more, some of which are very similar to the ones I listed (e.g. 1) is easily seen to be eqiuvalent to the existence of a left inverse).

Originally Posted by Kuma

If the matrix is not nxn, it is never invertible?

For we have the concepts of left and right inverse: is a left inverse of iff and a left-invertible matrix is a matrix with at least one left inverse. Similar considerations for right inverse.