Two matrices can only be added or subtracted if they have the same order. The operation is performed entry by entry.
$$A + B = (a_{ij} + b_{ij}) \qquad A - B = (a_{ij} - b_{ij})$$
$$A = \begin{pmatrix} 2 & 5 \ -1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 1 \ 6 & -2 \end{pmatrix}$$
$$A + B = \begin{pmatrix} 2+4 & 5+1 \ -1+6 & 3+(-2) \end{pmatrix} = \begin{pmatrix} 6 & 6 \ 5 & 1 \end{pmatrix}$$
$$A - B = \begin{pmatrix} 2-4 & 5-1 \ -1-6 & 3-(-2) \end{pmatrix} = \begin{pmatrix} -2 & 4 \ -7 & 5 \end{pmatrix}$$
Multiplying a matrix by a scalar (a single number $k$) multiplies every entry by that number.
$$kA = (k \cdot a_{ij})$$
$\$3 \times \begin{pmatrix} 1 & -2 \ 4 & 0 \end{pmatrix} = \begin{pmatrix} 3 & -6 \ 12 & 0 \end{pmatrix}$$
$$2A - 3B \quad \text{where} \quad A = \begin{pmatrix} 4 & 1 \ 2 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 3 \ 0 & 2 \end{pmatrix}$$
Step 1: $2A = \begin{pmatrix} 8 & 2 \ 4 & 10 \end{pmatrix}$
Step 2: $3B = \begin{pmatrix} 3 & 9 \ 0 & 6 \end{pmatrix}$
Step 3: $2A - 3B = \begin{pmatrix} 8-3 & 2-9 \ 4-0 & 10-6 \end{pmatrix} = \begin{pmatrix} 5 & -7 \ 4 & 4 \end{pmatrix}$
| Property | Rule |
|---|---|
| Commutative (addition) | $A + B = B + A$ |
| Associative (addition) | $(A+B)+C = A+(B+C)$ |
| Additive identity | $A + O = A$ |
| Scalar distributive | $k(A+B) = kA + kB$ |
Note: matrix multiplication is generally not commutative ($AB \neq BA$).
A school has two campuses with sales matrices (rows = item type, columns = term):
$$\text{North} = \begin{pmatrix} 120 & 95 \ 80 & 110 \end{pmatrix}, \quad \text{South} = \begin{pmatrix} 90 & 105 \ 70 & 85 \end{pmatrix}$$
Combined total: $\text{North} + \text{South} = \begin{pmatrix} 210 & 200 \ 150 & 195 \end{pmatrix}$
COMMON MISTAKE: Attempting to add matrices of different orders. Always check both matrices share identical order before adding or subtracting.
EXAM TIP: When combining $pA + qB$, apply scalar multiplication to each matrix first, then add. Show the intermediate steps for full marks.
