Scalar Multiplication of Matrices

In matrix algebra, a real number is called a scalar .

The scalar product of a real number, $r$ , and a matrix $A$ is the matrix $r A$ . Each element of matrix $r A$ is $r$ times its corresponding element in $A$ .

Given scalar $r$ and matrix $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}], r A = [\begin{matrix} r a_{11} & r a_{12} \\ r a_{21} & r a_{22} \end{matrix}]$ .

Example 1:

Let $A = [\begin{matrix} 2 & 1 \\ 3 & - 2 \end{matrix}]$ , find $4 A$ .

$4 A = 4 [\begin{matrix} 2 & 1 \\ 3 & - 2 \end{matrix}] = [\begin{matrix} 4 \cdot 2 & 4 \cdot 1 \\ 4 \cdot 3 & 4 \cdot (- 2) \end{matrix}] = [\begin{matrix} 8 & 4 \\ 12 & - 8 \end{matrix}]$

Properties of Scalar Multiplication:

Let $A$ and $B$ be $m \times n$ matrices. Let $O_{m \times n}$ be the $m \times n$ zero matrix and let $p$ and $q$ be scalars.

Properties of Scalar Multiplication
Associative Property	$p (q A) = (p q) A$
Closure Property	$p A$ is an $m \times n$ matrix.
Commutative Property	$p A = A p$
Distributive Property	$\begin{array}{l} (p + q) A = p A + q A \\ p (A + B) = p A + p B \end{array}$
Identity Property	$1 \cdot A = A$
Multiplicative Property of $- 1$	$(- 1) A = - A$
Multiplicative Property of $0$	$0 \cdot A = O_{m \times n}$

Scalar Multiplication of Matrices

Properties of Scalar Multiplication:

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