Addition and subtraction of matrices

Section 3.2 Addition and subtraction of matrices

Subsection 3.2.1 Definitions of addition and subtraction of matrices

For two matrices

$A=[a_{i,j}]$ and

$B = [b_{i,j}]\text{,}$ addition is defined if and only if the matrices have the same size. In that case, we say that the matrix

$C = [c_{i,j}]$ satisfies

$C=A+B$ if and only if

$\begin{equation*} c_{i,j} = a_{i,j}+b_{i,j} \end{equation*}$

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for all

$1\leq i\leq m$ and

$1\leq j\leq n\text{.}$

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Similarly, for two matrices

$A$ and

$B$ of the same size,

$C=A-B$ is defined by

$\begin{equation*} c_{i,j} = a_{i,j}-b_{i,j} \end{equation*}$

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for all

$1\leq i\leq m$ and

$1\leq j\leq n\text{.}$ When two matrices are of the same size, and hence their addition is defined, they are called conformable for addition.

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Example 3.2.1. Addition and subtraction of matrices.

$\begin{equation*} A= \begin{bmatrix} 1 \amp 2 \amp 3\\ 4 \amp 5 \amp 6 \end{bmatrix} \text{ and } B= \begin{bmatrix} 5 \amp 3 \amp 1\\ 0 \amp -1 \amp -2 \end{bmatrix} \end{equation*}$

then

$\begin{equation*} A+B= \begin{bmatrix} 6 \amp 5 \amp 4\\ 4 \amp 4 \amp 4 \end{bmatrix} \end{equation*}$

and

$\begin{equation*} A-B= \begin{bmatrix} -4 \amp -1 \amp 2\\ 4 \amp 6 \amp 8 \end{bmatrix} \end{equation*}$

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In short, addition and subtraction of two matrices are carried out by adding or subtracting the corresponding positions within the matrices.

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Subsection 3.2.2 Some properties of addition of matrices

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Theorem 3.2.2. Addition properties of Matrices.

Suppose $A\text{,}$ $B$ and $C$ are matrices of the same size, then

If $A$ and $B$ are $m\times n$ matrices, then so is $A+B\text{.}$
$A+B=B+A\qquad$ (commutativity of addition)
$(A+B)+C = A+(B+C)\qquad$ (associativity of addition)

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Proof.

By the definition of matrix addition, the sum of two matrices is a matrix of the same size.
We use $A=[a_{i,j}]$ and $B=[b_{i,j}]\text{.}$ The $i$-$j$ entry of $A+B$ is $a_{i,j}+b_{i,j}$ while the $i$-$j$ entry of $B+A$ is $b_{i,j}+a_{i,j}\text{.}$ Hence $A+B=B+A$ means $a_{i,j}+b_{i,j}=b_{i,j}+a_{i,j}$ for each possible $i$ and $j\text{.}$ We know this latter equation is valid since it uses the known commutative property of real numbers. (see properties of real numbers in Subsection 8.1.2.)
The $i$-$j$ entries of $(A+B)+C$ and $A+(B+C)$ must be equal. This says $(a_{i,j}+b_{i,j})+c_{i,j}=a_{i,j}+(b_{i,j}+c_{i,j})$ for all possible $i$ and $j\text{,}$ and this equation is valid by the distributive property of real numbers.