Re: linear algebra identities of inverse matricies/transpose

Left Inversion in Rectangular Cases: A subscript(left) = Al

Let Al^(-1) = (A^T * A)^(-1) * A^T

show Al^(-1) * A = I.

This matrix is called the left-inverse of A and it can be shown that if A in R^(mxn) such that A has a pivot in every column then the left inverse exists.

Right Inversion in Rectangular Cases. A subscript(right) = Ar

Let Ar^(-1) = A^T * (A * A^T)^(-1).

Show A * Ar^(-1) = I.

This matrix is called the right-inverse of A and it can be shown that if A in R^(mxn) such that A has a pivot in every row then the right inverse exists.

I tried the left part and this is what I did:

Al^(-1) / (A^T * A)^(-1) = A^T

Al^(-1) * (A^T * A) = A^T

Al^(-1) * A = A^T * (A^T)^(-1) = I

therefore, Al^(-1) * A = I

Can someone please help? I'm not sure if this is correct or not, so I want to see if I have the right idea. I know that A*A^(-1) = I so I thought this would work. Also isn't the right one the exact same thing? Or do I have to do that one a different way? Oh and can someone also explain the concept of left and right inverse. I don't really understand it. Thanks