1. ## Non-square inverse proofs

I need to prove the following:

1 A non-square matrix cannot have both a left and a right inverse
2 If a non-square matrix has a left(right) inverse, it has infinitely many.
3 If m<n, a non-square matrix has a right inverse if and only if rank A=m

2. Originally Posted by needhelp32
I need to prove the following:

1 A non-square matrix cannot have both a left and a right inverse
2 If a non-square matrix has a left(right) inverse, it has infinitely many.
3 If m<n, a non-square matrix has a right inverse if and only if rank A=m
It is always true that rank(AB)≤rank(B) (because the rows of AB are linear combinations of the rows of B). Also, rank(AB)≤rank(A) (because the columns of AB are linear combinations of the columns of A).

So for example if A is an m×n matrix with m<n, then rank(A)≤m, and rank(AB)≤m. Therefore AB can never be equal to the n×n identity matrix, which has rank n.

3. Thank you very much. I've now figured out (1). Does anyone know how to prove (2) in general.

4. Originally Posted by needhelp32
Does anyone know how to prove (2) in general.
If A is m×n with m<n, then rank(A) < n, so A has a nonzero kernel. Let x be a nonzero vector in this kernel, and let C be the matrix of any linear transformation whose range consists of multiples of x. Then AC=0. If B is a right inverse of A then so is B+λC, for any value of the scalar λ.