Row into column: why multiplication needs the inner orders to match

What to assume they know

• The order of a matrix, read rows first
• Addition and scaling of matrices, and the order conditions they carry
• The double subscript, and reading a whole row and a whole column out of an array
• Sigma notation for a finite sum, and what a summation index ranges over
• Arithmetic with signed numbers and simple algebraic expressions
• Reading a two-way table of quantities against a list of unit prices

What they should be able to do

• Explain where the sum of products in a matrix product comes from, using a quantities-against-prices situation
• State the condition under which a product is defined, in terms of the two orders
• Predict the order of a product without computing any entry
• Compute one named entry of a product by pairing a row against a column
• Compute a whole product of small matrices, keeping the intermediate sums visible
• Read and use the summation form of the entry definition, naming what the index runs over
• Decide, from orders alone, which of a list of matrix expressions are defined
• Handle products involving a single row or a single column, including one that collapses to a single number
• Interpret the entries of a product in an applied setting, saying what each one counts

Where it usually goes wrong

• "Multiply matrices the way you add them, position against position." That operation exists in mathematics and is not this one. An entry of this product is a sum over a whole row and a whole column, and nothing about the answer's entry at a position depends on the two factors' entries at that position alone.
• "The two matrices must have the same order." They must have matching inner counts. A two by three times a three by four is fine; a two by three times a two by three is not defined at all.
• "If the product exists one way it exists the other way." Not in general. The chapter states the exact condition for both to exist and it is two equations, not one. This is the hinge into the next topic.
• "The answer has the order of the bigger matrix." The answer takes the outer counts: rows from the left factor, columns from the right. Predict it before computing, every time.
• "A row times a column and a column times a row are the same thing." One collapses to a single entry, the other expands to a full square array. Exercise 3.2 Q3(ii) and Miscellaneous Exercise Q4 sit on opposite sides of this and both should be shown.
• "The summation formula is a new definition." It is the same sentence written with a sigma. Show the sum of n products written out and the sigma form on the same screen.
• "Ten dozen books is ten books." Exercise 3.2 Q20 is a unit-conversion trap wearing a matrix costume; the conversion happens before the product is formed.
• "An applied product's entries are just numbers." In the manufacturer item each entry is a revenue or a cost for a named market, and a student who cannot say which has not finished the question.

Questions to check understanding

• Decide whether a stated product is defined, from the two orders alone
• State the order of a product before computing it
• Compute one named entry of a product, showing the row and the column used
• Compute a full product of two small matrices of unequal shapes
• Write the entry rule in summation form and say what the index ranges over
• Choose the correct restriction on lettered orders so that a stated combination exists — the form of Exercise 3.2 Q21
• Evaluate a row times a matrix times a column and solve the resulting equation — the form of Miscellaneous Exercise Q4 and Q6
• Set up and evaluate a revenue or cost calculation as a matrix product, and say what each entry of the answer counts

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The shopping build-up (§3.4.5, Part I pp. 50–51). Two friends, two goods, a pen at five rupees and a story book at fifty. The first needs two pens and five books, the second eight pens and ten books. Verified: the first spends two hundred and sixty and the second five hundred and forty. The chapter then writes the requirement as a two by two matrix and the prices as a two by one column, and the two bills as a two by one column of unresolved sums. That unresolved column is the single most useful frame in the section — the sum of products is visible before it is evaluated.
• The second shop, and the second column (Part I pp. 50–51). Prices of four and forty at a second shop; verified, the two bills are two hundred and eight and four hundred and thirty-two. The chapter then sets the two price lists side by side as a two by two matrix and gets a two by two answer, and this is where the general shape appears: a second column of prices produces a second column of bills, and nothing else changes. Say that out loud; it is why the right factor's columns are handled independently.
• The definition (Part I p. 51). Stated in three passes: an observation in words about which counts must agree; then the formal statement, with the left factor of order m by n, the right of order n by p and the answer of order m by p; then the entry formula, written out as a sum of n products and again in summation form with the index running from one to n.
• The worked panel of four entries (Part I p. 52). A two by three matrix holding one, minus one and two above zero, three and four, times a three by two holding two and seven, minus one and one, five and minus four. The chapter computes the four entries in four separate displays, and in each one the relevant row of the left factor and the relevant column of the right factor are printed with a tinted background. Read this off the page image — the tinting is invisible in the text layer and it is the whole teaching device. Verified: the answer is thirteen and minus two above seventeen and minus thirteen.
• Example 12 (Part I p. 52). A two by two holding six and nine above two and three, times a two by three holding two, six and zero above seven, nine and eight. The chapter opens by checking the orders and only then computes. Verified: the answer is seventy-five, one hundred and seventeen and seventy-two above twenty-five, thirty-nine and twenty-four. Keep the chapter's ordering of the two steps — the check first, then the arithmetic.
• Exercise 3.2 Q1, parts (iv) and (v) (Part I p. 58). Two two by two matrices are given: two and four above three and two, and one and three above minus two and five. Verified: their product one way is minus six and twenty-six above minus one and nineteen; the other way it is eleven and ten above eleven and two. The two answers differ, which is the next topic's subject; here the item is just practice.
• Exercise 3.2 Q3 (Part I p. 58). Six products of assorted shapes. Verified: (i) gives a scalar matrix whose repeated value is the first letter squared plus the second letter squared. (ii) is a three by one against a one by three and gives a three by three holding two, three and four; four, six and eight; six, nine and twelve. (iii) gives a two by three holding minus three, minus four and one above eight, thirteen and nine. (iv) gives a three by three holding fourteen, zero and forty-two; eighteen, minus one and fifty-six; twenty-two, minus two and seventy. (v) gives a three by three holding one, two and three; one, four and five; minus two, two and zero. (vi) gives a two by two holding fourteen and minus six above four and five. Item (ii) is the one to show for section 10's neighbour — a column times a row expands, where a row times a column collapses, and students conflate the two constantly.
• Exercise 3.2 Q21 and Q22 (Part I p. 61). Two multiple-choice items on five matrices whose orders are given in letters. Verified: for the sum of two products to exist, the shared count must be three and the two outer counts must agree, so the first option is right; and for the combination in the second item the order is two by the second letter, so the second option is right. These two items are the whole of section 9 and they are the only place in the chapter where a student is asked to reason about orders with no numbers present at all.
• Exercise 3.2 Q19 and Q20 (Part I pp. 60–61). Q19 splits thirty thousand rupees between bonds paying five and seven per cent, for two target incomes. Verified: eighteen hundred needs fifteen thousand in each; two thousand needs five thousand in the first and twenty-five thousand in the second. Q20 totals a bookshop's takings from ten dozen, eight dozen and ten dozen books at eighty, sixty and forty rupees. Verified: twenty thousand one hundred and sixty rupees — and the dozens are the trap, since the counts must be multiplied by twelve before the product is formed.
• Miscellaneous Exercise Q4 and Q6 (Part I p. 72). Both are a row times a three by three times a column, equated to the zero matrix. Verified: Q4 gives the unknown as minus one; Q6 reduces to the square of the unknown less forty-eight, so the unknown is plus or minus four root three. Q6's answer is a pair and the question asks in the singular; say both values. These two items are section 10.
• Miscellaneous Exercise Q7 (Part I pp. 72–73). Three products sold in two markets, with unit prices and unit costs given. Verified: revenue is forty-six thousand in the first market and fifty-three thousand in the second; costs are thirty-one thousand and thirty-six thousand; so the gross profits are fifteen thousand and seventeen thousand, thirty-two thousand in all. This is the best applied item in the chapter because every entry of every intermediate product means something a shopkeeper would recognise, and section 11 should name each one as it appears.

Figures to have open

• An unresolved-sum column for section 1: two bills each written as a product plus a product, before evaluation. The chapter prints exactly this on Part I p. 50 and it is easy to skip past.
• A four-panel product computation for section 7 with one row of the left factor and one column of the right factor highlighted per panel. This is the chapter's own device on Part I p. 52, including the tinted highlighting. Build it with the repo's DataTable component.
• An orders strip for sections 3 and 4: the four counts in a line, with the inner pair joinable and the outer pair extractable, drawn once and reused. Not in the book.
• A collapse-and-expand pair for section 10: a row times a column shrinking to one entry, beside a column times a row opening into a square. Not in the book.
• No redraw of a textbook figure is possible or needed: this chapter prints no figure at all. Confirmed on the page image of every one of the forty-two pages.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 3 "Matrices", §3.4.5, the shopping build-up, Part I pp. 50–51
• The formal definition and the summation form, Part I p. 51
• The four-panel worked computation, Part I p. 52
• Example 12, Part I p. 52
• Exercise 3.2, questions 1(iv), 1(v), 3, 19, 20, 21 and 22, Part I pp. 58–61
• Miscellaneous Exercise on Chapter 3, questions 4, 6 and 7, Part I pp. 72–73
• Summary, the product bullet with its summation form, Part I p. 74