Sooji Shin

Theorem.  Let A be a p×q matrix and B be a q×p matrix, while p and q are positive integers satisfying p > q. Then the product matrix AB is not invertible.

Proof.  Note that an m×n matrix can be regarded as a linear transform from Rⁿ to R^m. Note further that the following statements are equivalent:

1. AB is invertible.
2. AB is a one-to-one linear transform from Rp onto itself.
3. ker(AB) = {0}.
4. dim(AB(R^p)) = p. (i.e. the dimension of range of AB is p.)

These are a fundamental theorem of Linear Algebra.
Thus we now only have to show that dim(AB(R^p)) ≠ p.

The dimension of row space of B is q, which equals the dimension of column space of B, thus the dimension of range of B cannot be larger than q, i.e. dim(B(R^p)) ≤ q. Besides, for the dimension of range of a linear transform cannot be larger than the dimension of domain, we have dim(A(B(R^p))) ≤ q. Since AB(R^p) = A(B(R^p)), we finally have dim(AB(R^p)) ≤ q < p. Proved by Sooji Shin.

Is there anyone who knows another proof?

Geschrieben von Sooji Shin