Suppose \(A\) is a \(2 \times 1\) matrix and that \(B\) is a \(1 \times 2\) matrix. Prove that \(C = AB\) is not invertible.
Solution
1. \(C = AB\) is a \(2 \times 2\) matrix of rank at most \(1\).
\(A\) has rank at most \(1\) (it is \(2 \times 1\)), and \(\operatorname{rank}(AB) \le \operatorname{rank}(A) \le 1\).
2. A \(2 \times 2\) matrix is invertible iff it has rank \(2\).
standard.
3. Hence \(C\) has rank \(\le 1 < 2\), so \(C\) is not invertible.
1 and 2.
4. Q.E.D.
3.