Matrix Scalar Product with Zero gives Zero Matrix
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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 |
$\blacksquare$
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices: $6$