Matrix Scalar Product with Zero gives Zero Matrix

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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.


\(\ds 0 \mathbf A\) \(=\) \(\ds 0 \sqbrk a_{m n}\) 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