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*}

for all \(1\leq i\leq m\) and \(1\leq j\leq n\text{.}\)

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*}

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.

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*}

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

Subsection 3.2.2 Some properties of addition of matrices

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)

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.