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.

Proof

 $\ds 0 \mathbf A$ $=$ $\ds 0 \sqbrk a_{m n}$ Definition of $\mathbf A$ $\ds$ $=$ $\ds \sqbrk {0 a}_{m n}$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \sqbrk 0_{m n}$ Zero is Zero Element for Multiplication $\ds$ $=$ $\ds \mathbf 0$ Definition of Zero Matrix

$\blacksquare$