Matrix multiplication: check row–column products, dimensions and units

Name every row and column first

The groups, pack types and counts are original KitNelo teaching material, not real course records. The single question is how many black-and-white and color pages each group needs after receiving specified pack counts. The unit is a printed page, not sheets after duplex layout, and excludes binding or waste pages.

A has Morning and Afternoon rows and X, Y, Z pack-count columns. B must retain that same X, Y, Z row order; its columns contain black-and-white and color pages per pack. A has two rows and three columns, B three rows and two columns. Preserve labels so the middle categories match in meaning as well as count.

Use one row and one column for each output

OpenStax’s matrix-operations chapter requires equal inner dimensions and gives the product the left matrix’s row count and the right matrix’s column count. Here (2×3)×(3×2) produces 2×2. Mark those four output positions before calculating.

For Morning’s black-and-white output, match row (2, 1, 0) with column (4, 2, 1): 2×4+1×2+0×1=10. Each term multiplies a pack count by pages per pack. Add the contributions for that one page category.

Expand the remaining cells fully. Zero explicitly means no packs of that type in this example; it is not missing data. Do not remove 0×5 and shift the later category into its position. The reason-for-each-step exercise helps check whether you can identify where every factor came from.

Check the totals through a different route

The cell answers should reconcile with a separate aggregation. Across both groups, X totals 2+1=3 packs, Y totals 1+0=1 and Z totals 0+3=3. Compute all printing by pack type next.

Black-and-white pages total 3×4+1×2+3×1=17; color pages total 3×1+1×5+3×2=14. These match the output-column sums 10+7 and 7+7. All printed pages total 17+14=31.

This check reconstructs contributions to a total; it does not require division by pack counts. The weighted-average contribution exercise distinguishes totals from averages. The deliverable here is total printing, so an extra final division would answer another question.

Why valid dimensions can still answer the wrong question

Matching dimensions permit the numerical operation; they do not align categories automatically. If B’s rows become Z, X, Y while A’s columns remain X, Y, Z, multiplying by position gives Morning 6 black-and-white and 5 color pages. The arithmetic executes but matches the wrong pack types. Repair the label alignment rather than editing the final values.

Order is also part of the question. BA is numerically defined here but produces 3×3, not two groups by two page types. B’s column labels are page categories while A’s row labels are groups. Both happen to have two positions, but they do not provide this task’s common middle category. Equal position counts alone do not preserve the original model.

Predict one more Z pack, then recompute

Change only Afternoon’s Z count from 3 to 4. One Z pack contains 1 black-and-white and 2 color pages, so predict additions of 1 and 2 to Afternoon’s outputs while Morning remains unchanged.

Full recomputation gives Afternoon black-and-white 1×4+0×2+4×1=8 and color 1×1+0×5+4×2=9. The new rows are (10, 7) and (8, 9), totaling 18 black-and-white and 16 color pages, or 34 pages altogether. The increase from 31 is 3, exactly one Z pack’s pages.

Explain the category matches through which that changed input traveled. This small model assumes fixed pages per pack and omits binding, copy waste and other operating constraints. Add those conditions for real printing instead of treating the exercise as a complete production plan.