Matrix Arithmetic

Matrix Addition, Subtraction, and Scalar Multiplication

Addition and Subtraction

Two matrices can only be added or subtracted if they have the same order. The operation is performed entry by entry.

\[A + B = (a_{ij} + b_{ij}) \qquad A - B = (a_{ij} - b_{ij})\]

Worked Example — Addition

\[A = \begin{pmatrix} 2 & 5 \\ -1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 1 \\ 6 & -2 \end{pmatrix}\]

\[A + B = \begin{pmatrix} 2+4 & 5+1 \\ -1+6 & 3+(-2) \end{pmatrix} = \begin{pmatrix} 6 & 6 \\ 5 & 1 \end{pmatrix}\]

Worked Example — Subtraction

\[A - B = \begin{pmatrix} 2-4 & 5-1 \\ -1-6 & 3-(-2) \end{pmatrix} = \begin{pmatrix} -2 & 4 \\ -7 & 5 \end{pmatrix}\]

Scalar Multiplication

Multiplying a matrix by a scalar (a single number $k$) multiplies every entry by that number.

\[kA = (k \cdot a_{ij})\]

Worked Example

$\$3 \times \begin{pmatrix} 1 & -2 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 3 & -6 \\ 12 & 0 \end{pmatrix}$\$

Combining Operations

\[2A - 3B \quad \text{where} \quad A = \begin{pmatrix} 4 & 1 \\ 2 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 3 \\ 0 & 2 \end{pmatrix}\]

Step 1: $2A = \begin{pmatrix} 8 & 2 \\ 4 & 10 \end{pmatrix}$

Step 2: $3B = \begin{pmatrix} 3 & 9 \\ 0 & 6 \end{pmatrix}$

Step 3: $2A - 3B = \begin{pmatrix} 8-3 & 2-9 \\ 4-0 & 10-6 \end{pmatrix} = \begin{pmatrix} 5 & -7 \\ 4 & 4 \end{pmatrix}$

Properties

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 ($AB \neq BA$).

Practical Context

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.

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