Let n N Let A be a leftinvertible n times nmatrix Prove that

Let n N. Let A be a left-invertible n times n-matrix. Prove that A is invertible.

Solution

Given that the square matrix A of order nxn is left invertible. We need to show that matrix A is invertible.

This can be easily done by first showing that if a square matrix is left invertible then it must be right invertible too. And we know that if a square matrix is both left and right invertible, then it is an invertible matrix.

In order to show that matrix A is right invertible, let us start with assuming that B is the left inverse of A.

If B is the left inverse of A, then we can write that BA = I. Now, by transpose property of matrix multiplication, we can write:

(BA)^T = A^TB^T

We know that BA=I, therefore, we get:

I^T = A^TB^T

A^TB^T = I^T = I

A^TB^T = I

Therefore, B^T is right inverse of A^T. And thus, B^T must be the left inverse of A^T too.

Hence, (AB)^T = B^TA^T = I

Therefore, AB = I

Therefore, B is right inverse of A too.

Since B is both left and right inverse of the square matrix A. Therefore, B is the inverse of matrix A.

Hence a left invertible square matrix is an invertible matrix.