Matrix Scalar Product Distributes over Matrix Entrywise Addition

Theorem

Let $\mathbf A$ and $\mathbf B$ be matrices both of order $m \times n$.

Let $k$ be a scalar.

Then:

$k \paren {\mathbf A + \mathbf B} = k \mathbf A + k \mathbf B$

where:

$k \mathbf A$ etc. denotes matrix scalar product.

Let $a_{i j}$ and $b_{i j}$ denote the $\tuple {i, j}$th entry in $\mathbf A$ and $\mathbf B$ respectively.

$\ds $

$\ds k \mathbf A + k \mathbf B$

$\ds \forall i \in \closedint 1 m, \forall j \in \closedint 1 n: \, $

$\ds $

$=$

$\ds k a_{i j} + k b_{i j}$

$\ds $

$=$

$\ds k \paren {a_{i j} + b_{i j} }$

$\ds $

$=$

$\ds k \paren {\mathbf A + \mathbf B}$

$\blacksquare$