Prove or disprove Suppose A B and C are all nxn matrices s

Prove or disprove: Suppose A , B , and C are all nxn matrices such that C =AB.

If C is invertible, then A is invertible. You may use the fact that if MxN = I , then NxM = I for any nxn matrices M and N

Solution

We are given that C = AB and that C is invertible. If so, then AB is also invertible (as AB = C) Then, det(AB) 0. However, det(AB) = detA detB so that detA detB 0, so both det A and det B are non-zero, and therefore both A and B are invertible.

NOTE:

We have used the property of invertible matrices that “ A square matrix A is invertible if and only if det(A) is not 0”.