# Answer to Question #23764 in Linear Algebra for Matthew Lind

Question #23764

Let A be an M by N matrix. When does A have a left inverse? When does it have a right inverse?

Expert's answer

Asrank(AB)<=max{rank(A),rank(B)}, then there is such N by M matrix B

that BA=1_n (identity matrix of size N) iff

n=rank(1_n)<=min{rank(A),rank(B)}.

If M<N then N<=rank(A)<M and so there is no left inverse.

If M>N then only possible case is rank(A)=rank(B)=N.

So, M by N matrix A has left inverse iff N<M and rank(A)=N.

So, M by N matrix A has right inverse iff N>M and rank(A)=M.

that BA=1_n (identity matrix of size N) iff

n=rank(1_n)<=min{rank(A),rank(B)}.

If M<N then N<=rank(A)<M and so there is no left inverse.

If M>N then only possible case is rank(A)=rank(B)=N.

So, M by N matrix A has left inverse iff N<M and rank(A)=N.

So, M by N matrix A has right inverse iff N>M and rank(A)=M.

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