Home page for accesible maths 1 Matrices 1.1 Motivation and definitions 1.3 Vector multiplication

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1.2. Matrix addition and scalar multiplication

Now that we have defined matrices, we will define their arithmetic operations. We start with the addition and scalar multiplication; they are the “obvious” operations. Let $n,m\geq 1$ be integers.

Definition 1.2.1.

Let $A=(a_{ij})\;,\;B=(b_{ij})\in\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ , and let $\lambda\in{\mathbb{R}}$ . We define:

(i)

The sum of $A$ and $B$ is the matrix $A+B\in\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ whose $(i,j)$ coefficient is $a_{ij}+b_{ij}$ .
(ii)

The scalar multiplication of $A$ by $\lambda$ (a scalar) is the matrix $\lambda A\in\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ whose $(i,j)$ coefficient is $\lambda a_{ij}$ .

Remark 1.2.2.

The sum of two matrices $A$ and $B$ is only defined when $A$ and $B$ have the same size, in which case the sum also has the same size.

Example 1.2.3.

Consider the matrices

$A=\begin{pmatrix}1&2&3\\ 4&5&6\\ \end{pmatrix}\quad\hbox{and}\quad B=\begin{pmatrix}-2&-2&-2\\ 0&1&-3\\ \end{pmatrix}\;.$

We calculate $2A$ and $A+B$ as follows

$2A=\begin{pmatrix}2&4&6\\ 8&10&12\end{pmatrix}\quad\hbox{and}\quad A+B=\begin{pmatrix}-1&0&1\\ 4&6&3\end{pmatrix}.$

Example 1.2.4. (Zero matrix)

Let ${{0}}_{n\times m}$ be the zero matrix in $\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ , that is the matrix with all zero coefficients. For instance

$0_{2\times 3}=\begin{pmatrix}0&0&0\\ 0&0&0\end{pmatrix}\in\operatorname{M}_{{2\times 3}}({{\mathbb{R}}}).$

Then, for any $A\in\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ :
1. (a)
  
  $0A={{0}}_{n\times m}$ , and
2. (b)
  
  (Additive inverse) $A+(-1)A={0}_{n\times m}$ .
For convenience, we will write $0$ for the zero matrix of any dimensions we want.