# Matrix Scalar Product is Associative

## 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 $\mathbf A$.

The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense:

For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$:

$\lambda \paren {\mu \mathbf A} = \paren {\lambda \mu} \mathbf A$

## Proof

 $\ds \lambda \paren {\mu \mathbf A}$ $=$ $\ds \lambda \paren {\mu \sqbrk a_{m n} }$ Definition of $\mathbf A$ $\ds$ $=$ $\ds \lambda \paren {\sqbrk {\mu a}_{m n} }$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \sqbrk {\lambda \paren {\mu a} }_{m n}$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \sqbrk {\paren {\lambda \mu} a}_{m n}$ Associative Law of Multiplication $\ds$ $=$ $\ds \paren {\lambda \mu} \sqbrk a_{m n}$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \paren {\lambda \mu} \mathbf A$ Definition of $\mathbf A$

$\blacksquare$