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Given a matrix $A=(a_{ij})_{ij}\in \mathcal{M}_{m\times n}(\mathbb{K})$ and a scalar $\alpha\in\mathbb{K}$, we define their product as the $m \times n$ matrix

$$ \alpha A=A\alpha = (\alpha a_{ij})_{ij} = \left(\begin{array}{cccc} \alpha a_{11} & \alpha a_{12} & \cdots & \alpha a_{1n}\\ \alpha a_{21} & \alpha a_{22} & \cdots & \alpha a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ \alpha a_{m1} & \alpha a_{m2} & \cdots & \alpha a_{mn}\\ \end{array}\right) $$