For $c\in\mathbb F$ and $A\in\mathbb F^{m\times n}$, define $(cA)_{ij}=ca_{ij}$. Scalar multiplication preserves shape. Example: $-2\begin{bmatrix}1&-3\\0&2\end{bmatrix}=\begin{bmatrix}-2&6\\0&-4\end{bmatrix}$. It obeys $1A=A$, $(cd)A=c(dA)$, and both distributive laws.
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