Matrix Scalar Product with Zero gives Zero Matrix

Theorem
Let $\Bbb F$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.

For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$.

When $\lambda = 0$, we have for all $\mathbf A$ in $\map \MM {m, n}$:


 * $0 \mathbf A = \mathbf 0$

where $\mathbf 0$ denotes the zero matrix.