The matrix scalar multiplication verifies the following properties

1. Distributivity with respect to scalars: $(\alpha+\beta)A=\alpha A+\beta A\qquad\forall\,\alpha,\beta\in\mathbb{K},\,A\in \mathcal{M}_{m\times n}(\mathbb{K})$
2. Distributivity with respect to matrices: $\alpha(A+B)=\alpha A+\alpha B\qquad\forall\,\alpha\in\mathbb{K},\,A,B\in \mathcal{M}_{m\times n}(\mathbb{K})$
3. Pseudoassociativity: $(\alpha\beta)A=\alpha(\beta A)\qquad\forall\,\alpha,\beta\in\mathbb{K},\,A\in \mathcal{M}_{m\times n}(\mathbb{K})$
4. Identity Property: $1A=A\qquad\forall\,A\in\mathcal{M}_{m\times n}(\mathbb{K})$

Let $\alpha$ and $\beta$ be two scalars in $\mathbb{K}$ and $A=(a_{ij})_{ij}$ and $B=(b_{ij})_{ij}$ be two matrices in $\mathcal{M}_{m\times n}(\mathbb{K})$

• Distributivity with respect to scalars: $$\begin{split} (\alpha+\beta)A & = ([\alpha+\beta] a_{ij})_{ij} \\ & = (\alpha a_{ij}+\beta a_{ij})_{ij} \\ & = (\alpha a_{ij})_{ij}+(\beta a_{ij})_{ij} \\ & = \alpha A+\beta A \end{split}$$
• Distributivity with respect to matrices: $$\begin{split} \alpha(A+B) & = \alpha[(a_{ij})_{ij}+(b_{ij})_{ij}] \\ & = \alpha[(a_{ij}+b_{ij})_{ij}] \\ & = (\alpha[a_{ij}+b_{ij}])_{ij} \\ & = (\alpha a_{ij}+\alpha b_{ij}])_{ij} \\ & = (\alpha a_{ij})_{ij}+(\alpha b_{ij})_{ij} \\ & = \alpha A+\alpha B \end{split}$$
• Pseudoassociativity: It is not called associativity because where dealing with elements of different sets ($\mathbb{K}$ and $\mathcal{M}_{m\times n}(\mathbb{K})$), but besides this we may think of it as associativity $$\begin{split} (\alpha\beta)A & = ([\alpha\beta] a_{ij})_{ij} \\ & = (\alpha[\beta a_{ij}])_{ij} \\ & = \alpha(\beta a_{ij})_{ij} \\ & = \alpha(\beta A) \end{split}$$
• Identity Property: $$\begin{split} 1A & = 1(a_{ij})_{ij} \\ & = (1a_{ij})_{ij} \\ & = (a_{ij})_{ij} \\ & = A \end{split}$$