Use matrix multiplication to answer an application question

Define what every row and column represents

Two sites order small and large kits. Quantity matrix Q has rows North,South and columns Small,Large. Cost matrix p has rows Small,Large and one column of dollars per kit. Shared categories must appear in the same order before multiplying

Matrix Q has rows North then South and columns Small then Large. How many large kits does the South site order?

The shared category disappears from the result

Qp has rows North,South and one dollar-total column. For each site, multiply the number of each kit by its cost and add. North costs 50 dollars and South costs 60 dollars. Multiplying kits by dollars per kit gives dollars

Given matrix

\(\displaystyle \begin{gathered}Q=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\\p=\begin{bmatrix} 8 \\ 13 \end{bmatrix}\end{gathered}\)

Quantity matrix Q has rows North then South and columns Small then Large. Cost matrix p has rows Small then Large, in dollars per kit. What is the combined cost in dollars for the two sites?

Different output columns can carry different units

Matrix P has rows Small then Large, and columns dollars per kit and minutes to assemble one kit. The product QP has rows North then South; its first column is dollars and its second is minutes. Do not add dollars to minutes merely because the values share a row

Given matrix

\(\displaystyle \begin{gathered}Q=\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}\\P=\begin{bmatrix} 8 & 2 \\ 13 & 5 \end{bmatrix}\end{gathered}\)

Quantity matrix Q has rows North then South and columns Small then Large. Matrix P has rows Small then Large and columns dollars per kit then assembly minutes per kit. How many assembly minutes does the South site require?

What must be checked before computing a contextual matrix product?

Build and challenge a model, ungraded

North orders 3 small and 2 large kits; South orders 1 small and 4 large kits. A small kit costs 8 dollars and takes 2 assembly minutes; a large kit costs 13 dollars and takes 5 assembly minutes. Q has rows North then South and columns Small then Large. P has rows Small then Large and columns dollars per kit then assembly minutes per kit. Construct Q and P from these data. Explain every output entry, then assess whether the cost and labour model remains valid with a bulk discount and one shared setup time. Suggest a correction and a way to check it against observations. Independent modelling and validation are ungraded. Before computing, choose an application and its matrix or spreadsheet functionality. Justify the choice and one limitation. Run it and retain the labelled inputs, formulas or commands, settings and actual outputs. Check one entry independently by a row-column dot product; submit your rationale and evidence for review

Model: linear contributions and exceptions

A matrix-capable application is suitable for calculating all entries of QP; a spreadsheet with labelled rows, columns and audit cells is another defensible choice. The calculation below independently checks the North assembly time in minutes. Neither tool establishes that the contextual assumptions hold. Q has rows North then South and columns Small then Large. P has rows Small then Large and columns dollars per kit then assembly minutes per kit. The matrix product assumes constant per-kit costs/times and additive contributions. It predicts site totals (50 dollars,16 minutes) and (60 dollars,22 minutes). A quantity-dependent discount needs an explicit pricing rule, so the fixed cost column may no longer apply. A single setup time must be added once at the correct aggregation level, not once per kit. Compare predicted totals with invoices and measured assembly times, identify systematic residuals, revise the assumptions and recheck new observations. The calculation itself cannot validate those assumptions. Keep the actual run record separate from these expected model totals