There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication. These form the basic techniques to work with matrices.

These techniques can be used in calculating sums, differences and products of information such as sodas that come in three different flavors: apple, orange, and strawberry and two different packaging: bottle and can. Two tables summarizing the total sales between last month and this month are written to illustrate the amounts. Matrix addition, subtraction and scalar multiplication can be used to find such things as: the sales of last month and the sales of this month, the average sales for each flavor and packaging of soda in the $2$-month period.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions.  Matrix addition is commutative and is also associative, so the following is true:

$\displaystyle A+B=B+A $

$\displaystyle (A+B)+C=A+(B+C)$

Adding matrices is very simple. Just add each element in the first matrix to the corresponding element in the second matrix.  

$\displaystyle \begin {pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}+\begin{pmatrix} 10 & 20 & 30 \\ 40 & 50 & 60 \end{pmatrix}=\begin {pmatrix} 11 & 22 & 33 \\ 44 & 55 & 66 \end {pmatrix}$

Note that element  in the first matrix, $1$, adds to element $x_{11}$ in the second matrix, $10$, to produce element $x_{11}$ in the resultant matrix, $11$. Also note that both matrices being added are $2\times 3$, and the resulting matrix is also $2\times 3$. You cannot add two matrices that have different dimensions.

As you might guess, subtracting works much the same way except that you subtract instead of adding.

$\displaystyle \begin{pmatrix} 10 & -20 & 30 \\ 40 & 50 & 60 \end{pmatrix}-\begin{pmatrix} 1 & -2 & 3 \\ 4 & -5 & 6 \end{pmatrix}=\begin{pmatrix} 9 & -18 & 27 \\ 36 & 55 & 54 \end{pmatrix}$

Once again, note that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have different dimensions. Be careful when subtracting with signed numbers.

Scalar Multiplication

In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. What does it mean to multiply a number by $3$? It means you add the number to itself $3$ times. Multiplying a matrix by $3$ means the same thing; you add the matrix to itself $3$ times, or simply multiply each element by that constant.

$\displaystyle 3\cdot \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}=\begin{pmatrix} 3 & 6 & 9 \\ 12 & 15 & 18 \end{pmatrix}$

The resulting matrix has the same dimensions as the original. Scalar multiplication has the following properties:

• Left and right distributivity: $(c+d)\textbf{M} = \textbf{M}(c+d) = \textbf{M}c+\textbf{M}d$
• Associativity: $(cd)\textbf{M} = c(d\textbf{M})$
• Identity: $1\textbf{M} = \textbf{M}$
• Null: $0\textbf{M} = \textbf{0}$
• Additive inverse: $(-1)\textbf{M} = -\textbf{M}$