Matrix operations (SACE Stage 2 General Mathematics)

Two stores hold stock as $S_1 = \begin{bmatrix} 10 & 4 \\ 6 & 8 \end{bmatrix}$ and $S_2 = \begin{bmatrix} 3 & 5 \\ 2 & 1 \end{bmatrix}$, where rows represent different products and columns represent sizes. Find the total combined stock, then find $2S_1$ to represent a planned doubled order for store 1.

Step 1: Check the matrices are conformable for addition

Before adding, we confirm both matrices have the same order. Both $S_1$ and $S_2$ are $2 \times 2$ matrices, so addition is defined. The result will also be $2 \times 2$.

Step 2: Add the matrices by combining matching entries

Matrix addition works element by element: each entry in the result comes from adding the entries in the same position of both matrices. We add the entry in row 1, column 1 of $S_1$ to the entry in row 1, column 1 of $S_2$, and so on for every position:

This result gives the total number of each product in each size across both stores.

Step 3: Apply scalar multiplication to find the doubled order

To double $S_1$, we multiply every entry in $S_1$ by the scalar 2. This scales the entire matrix uniformly, which makes sense here because we want twice as many of every item in the order:

Final answers: Total stock is $\begin{bmatrix} 13 & 9 \\ 8 & 9 \end{bmatrix}$; the doubled order for store 1 is $2S_1 = \begin{bmatrix} 20 & 8 \\ 12 & 16 \end{bmatrix}$.

A cafe order is $Q = \begin{bmatrix} 3 & 2 \end{bmatrix}$ (3 coffees, 2 muffins). Prices are $P = \begin{bmatrix} 4.50 \\ 3.00 \end{bmatrix}$ dollars. Find the total cost of the order using matrix multiplication, and state the order of the resulting matrix.

Step 1: Check that the product $QP$ is defined

Before multiplying, we confirm conformability. $Q$ has order $1 \times 2$ (one row, two columns) and $P$ has order $2 \times 1$ (two rows, one column). The inner dimensions are both 2, so the product is defined. The outer dimensions tell us the result will have order $1 \times 1$, that is, a single number, which makes sense for a total cost.

Step 2: Apply the row-by-column rule to compute $QP$

To find the single entry in the $1 \times 1$ product, we pair each element of row 1 of $Q$ with the matching element of column 1 of $P$ and sum the products. This is why matrix multiplication is suited to "quantity times price" problems: it automatically multiplies each quantity by its price and adds them:

Step 3: Interpret the result in context

The resulting $1 \times 1$ matrix holds the single value $19.50$. This represents the total cost of the order: 3 coffees at $\$4.50$ each plus 2 muffins at $\$3.00$ each comes to $\$19.50$.

Final answer: $QP = [19.50]$, so the total cost of the order is $\$19.50$.