Question: $\mathbf{P} = \begin{pmatrix} 5 & 3 & 2 \\ 7 & 4 & y \end{pmatrix}$ holds the labour hours, wood blocks and paint tins for a dining and a rocking chair; $\mathbf{Q} = \begin{pmatrix} 12 \\ x \\ 3 \end{pmatrix}$ holds the unit costs (labour $12/h, wood \

x$, paint \$3). Given $\mathbf{PQ} = \begin{pmatrix} 90 \\ 134 \end{pmatrix}$, find $x$ and $y$; then evaluate $\mathbf{SR}$ for $\mathbf{S} = \begin{pmatrix} 500 & 850 \end{pmatrix}$, $\mathbf{R} = \mathbf{PQ}$, and say what it represents. **Step 1.** Expand $\mathbf{PQ}$ row-by-column and equate entries:

60 + 3x + 6 = 90 ;\Rightarrow; 3x = 24 ;\Rightarrow; x = 8

**Step 2.** Second row, using $x = 8$:

84 + 4(8) + 3y = 134 ;\Rightarrow; 3y = 18 ;\Rightarrow; y = 6

**Step 3.** $\mathbf{R}$'s entries are the **cost per chair** (\$90 dining, \$134 rocking). Then

\mathbf{SR} = \begin{pmatrix} 500 & 850 \end{pmatrix}\begin{pmatrix} 90 \ 134 \end{pmatrix} = (500)(90) + (850)(134) = (158,900)

**Step 4.** Interpretation, in a sentence: $\usd{}158\,900$ is the **total cost of manufacturing** $500$ dining chairs and $850$ rocking chairs. **⚠ Watch out:** "Explain what your answer represents" wants the real-world sentence with units — not "it is S times R". Keep the ordering meaningful: quantities-row × cost-column works because the inner dimensions pair each quantity with its own unit cost.

Question: Evaluate $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \begin{pmatrix} 0 \\ 3 \\ -1 \end{pmatrix}$.

Step 1. Check compatibility first, by writing the orders side by side:

$$(2 \times \underline{3}) \; (\underline{3} \times 1)$$

The inner numbers match ($3 = 3$), so the product exists — and the outer numbers say the answer is a $2 \times 1$ matrix.

Step 2. Top entry: first row of the left matrix sweeps down the column of the right, multiply-and-add:

$$1(0) + 2(3) + 3(-1) = 0 + 6 - 3 = 3$$

Step 3. Bottom entry: second row, same column:

$$4(0) + 5(3) + 6(-1) = 0 + 15 - 6 = 9$$

Step 4. Assemble:

$$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \begin{pmatrix} 0 \\ 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \\ 9 \end{pmatrix}$$

⚠ Watch out: Matrix multiplication is row-by-column, never element-by-element — and it is not commutative: reversing the order here ($3\times1$ times $2\times3$) doesn't even exist. Run the inner-number check before computing anything; it also predicts the answer's shape for free.

Question: A bookstore's stock: Malay — 20 fiction, 32 textbooks, 30 general, 9 reference; Chinese — 31, 45, 38, 10; English — 80, 85, 73, 59. Represent this as a matrix, state its order, and write the row matrix for Malay books and the column matrix for fiction.

Step 1. Rows = languages, columns = book types, keeping the table's layout:

$$\begin{pmatrix} 20 & 32 & 30 & 9 \\ 31 & 45 & 38 & 10 \\ 80 & 85 & 73 & 59 \end{pmatrix}$$

Step 2. The order is rows × columns:

$$3 \times 4$$

Step 3. Malay books are the first row — a $1 \times 4$ row matrix:

$$\begin{pmatrix} 20 & 32 & 30 & 9 \end{pmatrix}$$

Step 4. Fiction is the first column — a $3 \times 1$ column matrix:

$$\begin{pmatrix} 20 \\ 31 \\ 80 \end{pmatrix}$$

Once data is in matrix form, whole-table operations come free: adding a second month's stock matrix element-by-element, or doubling stock as a scalar multiple $2\mathbf{A}$.

⚠ Watch out: Order is rows first, columns second — a $3 \times 4$ is not a $4 \times 3$, and every later question (especially multiplication) depends on this. Keep the row/column meanings fixed once chosen; swapping midway scrambles the data silently.