Non-square inverse proofs

• Nov 3rd 2007, 07:32 PM
needhelp32
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
• Nov 5th 2007, 01:40 AM
Opalg
Quote:

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.
• Nov 5th 2007, 11:01 AM
needhelp32
Thank you very much. I've now figured out (1). Does anyone know how to prove (2) in general.
• Nov 5th 2007, 12:00 PM
Opalg
Quote:

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.