Addition and Scalar Multiplication

Addition: entrywise, same order only

Definition. For two matrices of the same order $A = [a_{ij}]_{m \times n}$ and $B = [b_{ij}]_{m \times n}$, the sum is $$A + B = [a_{ij} + b_{ij}]_{m \times n}$$ — add corresponding entries. If the orders differ, $A + B$ is simply not defined.

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

Scalar multiplication: every entry scales

Definition. For a scalar $k$, $\ kA = [k\,a_{ij}]_{m \times n}$ — multiply every entry by $k$.

In particular $(-1)A = -A$ is the negative of $A$, and subtraction is $A - B = A + (-B)$: entrywise difference.

Worked combination. For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}$: $$2A - B = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix} - \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ 7 & 6 \end{pmatrix}$$

The algebra these operations satisfy

For matrices $A, B, C$ of the same order and scalars $k, l$:

Every rule is inherited entry-by-entry from ordinary number arithmetic — which is why this part of matrix algebra holds no surprises. (The surprises are saved for multiplication, next section.)

Common mistakes to avoid

Mistake 1 — adding different orders. A $2 \times 3$ plus a $3 \times 2$ is undefined, full stop.

Mistake 2 — scaling only one row. $kA$ multiplies every entry — scaling a single row is a different (elementary) operation, not scalar multiplication.

Mistake 3 — sign slips in $A - 2B$: distribute the $-2$ to every entry of $B$ before adding.

Mistake 4 — treating $O$ as "the" zero matrix: the additive identity must have the same order as $A$.

Solving Matrix Equations with $+$ and Scalars

The linear-equation template

Because addition and scalar multiplication obey familiar algebra, simple matrix equations solve exactly like linear equations in one unknown — with matrices in place of numbers.

Template 1 — isolate $X$: from $2A + 3X = 5B$, $$3X = 5B - 2A \implies X = \frac{1}{3}(5B - 2A)$$ compute $5B - 2A$ entrywise, then divide every entry by 3.

Template 2 — the sum-and-difference pair: given $X + Y = S$ and $X - Y = D$, $$X = \frac{1}{2}(S + D), \qquad Y = \frac{1}{2}(S - D)$$

Worked pair. $X + Y = \begin{pmatrix} 5 & 2 \\ 0 & 9 \end{pmatrix}$, $\ X - Y = \begin{pmatrix} 3 & 6 \\ 0 & -1 \end{pmatrix}$: $$X = \frac{1}{2}\begin{pmatrix} 8 & 8 \\ 0 & 8 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 0 & 4 \end{pmatrix}, \qquad Y = \frac{1}{2}\begin{pmatrix} 2 & -4 \\ 0 & 10 \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}$$

Template 3 — unknowns inside entries: a matrix equation with variables unpacks (by equality of matrices) into scalar equations. From $$2\begin{pmatrix} x & 5 \\ 7 & y - 3 \end{pmatrix} + \begin{pmatrix} 3 & -4 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 7 & 6 \\ 15 & 14 \end{pmatrix}$$ the $(1,1)$ entry gives $2x + 3 = 7 \Rightarrow x = 2$, and the $(2,2)$ entry gives $2(y - 3) + 2 = 14 \Rightarrow y = 9$ (the other two entries check out: $10 - 4 = 6$ ✓, $14 + 1 = 15$ ✓).

Data applications

Scalar multiplication scales tabulated data wholesale (doubling a production matrix), and addition combines tables (two months' sales). If a farmer's September and October production matrices are $A$ and $B$, then $A + B$ is the two-month total and $A - B$ the month-on-month change — entry by entry, with each entry keeping its row-column meaning.

Common mistakes to avoid

Mistake 1 — halving only one matrix in Template 2: both $S + D$ and $S - D$ get the $\frac{1}{2}$.

Mistake 2 — forgetting to verify the spare entries in Template 3: the unpacked system is overdetermined; a full-marks answer notes that every entry is satisfied.

Mistake 3 — moving a matrix across '=' without flipping sign — the algebra is familiar, but each move still applies to whole matrices.

Solved Examples

Example 1 — A straight sum

Find $A + B$ for $A = \begin{pmatrix} \sqrt 3 & 1 & -1 \\ 2 & 3 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & \sqrt 5 & 1 \\ -2 & 3 & \frac{1}{2} \end{pmatrix}$.

Step 1 — same order ($2 \times 3$) ✓, add entrywise: $$A + B = \begin{pmatrix} \sqrt 3 + 2 & 1 + \sqrt 5 & 0 \\ 0 & 6 & \frac{1}{2} \end{pmatrix}$$

Answer: as above — surds add formally; nothing simplifies unless the surds match.

Example 2 — A scalar combination

If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}$, find $2A - B$.

Step 1 — scale: $2A = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}$.

Step 2 — subtract entrywise: $$2A - B = \begin{pmatrix} 2 - 3 & 4 - 1 \\ 6 + 1 & 8 - 2 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ 7 & 6 \end{pmatrix}$$

Answer: $\begin{pmatrix} -1 & 3 \\ 7 & 6 \end{pmatrix}$.

Example 3 — Isolating the unknown matrix

Find the matrix $X$ such that $2A + 3X = 5B$, where $A = \begin{pmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{pmatrix}$.

Step 1 — isolate: $3X = 5B - 2A$.

Step 2 — compute the right side entrywise: $$5B - 2A = \begin{pmatrix} 10 - 16 & -10 - 0 \\ 20 - 8 & 10 + 4 \\ -25 - 6 & 5 - 12 \end{pmatrix} = \begin{pmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{pmatrix}$$

Step 3 — divide by 3: $$X = \begin{pmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{pmatrix}$$

Answer: as above — fractional entries are perfectly normal; do not force integers.

Example 4 — The sum-and-difference pair

Find $X$ and $Y$ if $X + Y = \begin{pmatrix} 5 & 2 \\ 0 & 9 \end{pmatrix}$ and $X - Y = \begin{pmatrix} 3 & 6 \\ 0 & -1 \end{pmatrix}$.

Step 1 — add the equations: $2X = \begin{pmatrix} 8 & 8 \\ 0 & 8 \end{pmatrix}$, so $X = \begin{pmatrix} 4 & 4 \\ 0 & 4 \end{pmatrix}$.

Step 2 — subtract the equations: $2Y = \begin{pmatrix} 2 & -4 \\ 0 & 10 \end{pmatrix}$, so $Y = \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}$.

Answer: $X = \begin{pmatrix} 4 & 4 \\ 0 & 4 \end{pmatrix}$, $\ Y = \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}$ (check: their sum and difference reproduce the data ✓).

Example 5 — Unknowns inside entries

Find $x$ and $y$ from $2\begin{pmatrix} x & 5 \\ 7 & y - 3 \end{pmatrix} + \begin{pmatrix} 3 & -4 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 7 & 6 \\ 15 & 14 \end{pmatrix}$.

Step 1 — combine the left side: $$\begin{pmatrix} 2x + 3 & 6 \\ 15 & 2y - 4 \end{pmatrix} = \begin{pmatrix} 7 & 6 \\ 15 & 14 \end{pmatrix}$$

Step 2 — equate the entries containing unknowns: $2x + 3 = 7 \Rightarrow x = 2$; $\ 2y - 4 = 14 \Rightarrow y = 9$.

Step 3 — confirm the spare entries already match ($6 = 6$, $15 = 15$) ✓.

Answer: $x = 2, \ y = 9$.

Example 6 — Combining data tables

A farmer's production (in tonnes) of rice and wheat over two plots is $A = \begin{pmatrix} 10 & 20 \\ 30 & 15 \end{pmatrix}$ in September and $B = \begin{pmatrix} 5 & 10 \\ 20 & 10 \end{pmatrix}$ in October (rows = plots, columns = crops). Find the total production and the decrease from September to October.

Step 1 — total: $A + B = \begin{pmatrix} 15 & 30 \\ 50 & 25 \end{pmatrix}$.

Step 2 — decrease: $A - B = \begin{pmatrix} 5 & 10 \\ 10 & 5 \end{pmatrix}$.

Answer: the entry conventions ride along: e.g. plot 2 produced 50 tonnes of rice in total, and every crop-plot combination decreased.

Example 7 — Scaling a data table

If each entry of the October matrix $B$ above doubles next October, write the new matrix.

Step 1 — scalar multiply: $$2B = \begin{pmatrix} 10 & 20 \\ 40 & 20 \end{pmatrix}$$

Answer: $2B$ — scalar multiplication is uniform scaling of the whole table, which is exactly why it models across-the-board percentage changes.