Module 12 — Matrix Operations
Use this if matrix sizes, addition, scalar multiplication, and matrix multiplication are mixing together.
Addition requires matching dimensions; multiplication requires the inner dimensions to agree.
Scalar multiplication scales every entry; matrix multiplication computes rows times columns.
In a product, rows and columns often represent different entities, not interchangeable numbers.
When can two matrices be added?
Only when they have the same dimensions.
Got it now? Scroll back up and re-read the section — it should land differently.
| Assignment | Textbook section | Practice focus |
|---|---|---|
| Matrix Add | §5.1 | Dimensions, entries, equality, addition and scalar combinations |
| Matrix Multi | §5.2 | Compatible dimensions, inner products, full products and applications |
A matrix with 2 rows and 3 columns has size 2 × 3. The entry a₂₃ sits in row 2, column 3. Count rows first, then columns.
Two matrices are equal when their sizes match and every corresponding entry agrees. For example, if the top-left entries are x + 2 and 9, then x + 2 = 9.
Add or subtract corresponding positions only when both sizes match. Scalar multiplication applies to every entry. A − B means subtract every entry of B, including negative entries.
Original worked example:
If rows are stores and columns are products, keep those labels in the same order before combining inventories.
For an m × n matrix times an n × p matrix, the inner sizes n must agree; the result has size m × p. Entry (i, j) uses row i of the first matrix and column j of the second. Multiply matching positions, then add the products.
For a whole product, repeat this calculation for every output position. Matrix multiplication does not generally give AB = BA; the reverse order may even be undefined.
Original example: rows of Q represent two market stalls; columns represent posters and notebooks.
Each entry of Qp is one stall’s revenue in dollars: quantities × matching prices, summed across products. Reversing the price order would change the meaning and the answer. For technology checks, enter each row carefully, verify dimensions, and compare one hand-computed entry with the output.
Use the Practice quiz for fresh problems, then Explain-it for two short written checks. Work on paper first. The practice feedback explains the method; it is not a WebAssign answer key.
