# Is there any 2x3 real matrix having a right and a left inverse?

I was thinking about this question like 1 hour, because the question not says that 2x3 matrix is invertible.

So I thought;

For right inverse of the 2x3 matrix, the product of them will be equal to 2x2 identity matrix.

For left inverse of the 2x3 matrix, the product of them will be equal to 3x3 identity matrix.

Since the question does not says that 2x3 matrix is invertible, identity matrices must not be equal to eachother either the inverses, so that there might be a 2x3 matrix that has a left and right inverse different from eachother?

If a matrix is invertible that means the inverse is unique, but since the question not saying this 2x3 matrix is invertible, I can't stop thinking that those inverses might be exist.

How can I prove that those are exist or not?

• A square matrix is eventually invertible, a non square matrix is never invertible. The pseudoinverses that you can find are non unique (you can have more than one left or right inverse) nor equal. – N74 Nov 1 '17 at 18:41
• @N74 so you are saying that it is possible to find a right and left inverse of a 2x3 matrix? – GLHF Nov 1 '17 at 18:44
• – N74 Nov 1 '17 at 18:47

No. If $A$ is a 2x3 matrix then rank of $A$ is at most 2 so, if $B$ is a 3x2 matrix then the rank of $BA$ is at most 2, then it cannot be equal to $I$.