When does the matrix $R$ fail to be invertible in the $QR$ decomposition?

Question

In my class, we say that the $QR$ decomposition of a $n \times m$ matrix $A$ is when $A = QR$ and $R$ is invertible and upper triangular and $Q$ has the property that $Q^T Q = I$. I looked at other sources online, and other people do not require that the matrix $R$ is invertible. I found examples of square matrices where the $R$ matrix failed to be invertible. Is there an example of a rectangular matrix where $R$ fails to be invertible?

Answer 1

Let me give you a hint: start with a rectangular matrix that is already upper triangular but is not invertible.