What multiplication keeps from ordinary algebra, and the two things it loses

What to assume they know

• The definition of a matrix product and the condition under which it exists
• The order of a product, read off the outer counts
• The identity matrix and the zero matrix, from the first module
• Equality of matrices, and reading one matrix equation as many scalar equations
• Adding and scaling matrices, and the laws they obey
• Proof by mathematical induction, from Class XI
• The compound-angle formulas for sine and cosine, from Class XI
• Solving a pair of simultaneous linear equations

What they should be able to do

• Show that two products taken in opposite orders need not agree, in the case where the two have different orders and in the case where they have the same order
• State the exact condition under which both products of a pair exist
• Produce a pair that does commute, and name a family in which commuting is guaranteed
• Produce two non-zero matrices whose product is the zero matrix, and say what inference this blocks
• State the associative law, the two distributive laws and the identity property for matrix multiplication
• Verify the associative law and a distributive law on stated triples
• Compute a power of a square matrix and evaluate a polynomial in that matrix
• Prove a formula for the nth power of a matrix by induction
• Solve for an unknown matrix that appears on a stated side of a product
• Say which ordinary-algebra manipulations remain legal and which do not

Where it usually goes wrong

• "Two matrix products are never equal." Some pairs commute, and the chapter prints one and names a whole family. The correct statement is that equality is not guaranteed, so it must be checked and may not be assumed.
• "If both products exist, they must have the same order." Only if both factors are square and of one order. Example 13 has both products existing at two by two and three by three.
• "A product being zero means one factor is zero." Example 15 refutes this in four entries. The inference students actually use it for is cancellation, and section 6 should name that consequence explicitly rather than leaving it as a curiosity.
• "You can cancel a common factor from both sides of a matrix equation." You cannot, and this chapter never does. Every solved matrix equation in the chapter is solved by rearranging with the laws that do hold.
• "Associativity failing would not matter much." It is what makes a power of a matrix well defined at all. Without it the cube in Example 18 would need a bracketing convention.
• "A polynomial in a matrix has an ordinary number as its constant term." It has a scalar multiple of the identity. Example 18 and Exercise 3.2 Q15, Q16 and Q17 all turn on this.
• "An unknown matrix can be moved to whichever side is convenient." It cannot, because the two sides are different operations. Miscellaneous Exercise Q8 and Example 25 both fix the side by the shape of the equation.
• "The properties in this section were proved." They were stated. The chapter says so in its own opening sentence and then verifies them on examples, which is a different thing.

Questions to check understanding

• Compute both products of a stated pair and show that they differ
• Produce a pair of two by two matrices whose two products agree
• Produce two non-zero matrices whose product vanishes
• Verify the associative law on a stated triple of matrices
• Verify a distributive law on a stated triple
• Evaluate a stated polynomial in a square matrix and show it vanishes — the form of Example 18 and of Exercise 3.2 Q16
• Find the scalar making a stated matrix identity hold — the form of Exercise 3.2 Q17
• Prove a formula for the nth power of a matrix by induction — the form of Miscellaneous Example 23
• Find an unknown matrix from a product equation, deducing its order first — the form of Miscellaneous Example 25 and Miscellaneous Exercise Q8
• Choose the correct option in an item that turns on a matrix equalling its own square — the form of Miscellaneous Exercise Q11

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 Remark on existence (Part I p. 53). It observes that one product may exist while the other does not, gives the reason using Example 12's pair, and then states the exact condition for both to exist as two equations on the four counts. It closes by noting that two square matrices of one order always have both. This Remark is the hinge of the topic and section 1 should open on it.
• Example 13 (Part I p. 53). A two by three holding one, minus two and three above minus four, two and five, times a three by two holding two and three; four and five; two and one. Verified: one way round the answer is two by two, holding zero and minus four above ten and three; the other way it is three by three, holding minus ten, two and twenty-one; minus sixteen, two and thirty-seven; minus two, minus two and eleven. The two answers cannot be equal because they are not the same size, which is the easy half of the failure.
• Example 14 (Part I p. 53). Two two by two matrices — one and zero above zero and minus one, and zero and one above one and zero. Verified: one product is zero and one above minus one and zero; the other is zero and minus one above one and zero. Same order, opposite signs off the diagonal. This is the example.
• The Note tempering the claim (Part I p. 54). One and zero above zero and two, against three and zero above zero and four. Verified: both products come to three and zero above zero and eight. The Note then remarks that diagonal matrices of a shared order always commute. Verified: the product of two diagonal matrices multiplies the diagonals position by position, and ordinary multiplication of numbers does not care about order. Section 4 exists because students over-learn the failure and start asserting that two products are never equal.
• Example 15 (Part I p. 54). Zero and minus one above zero and two, times three and five above zero and zero. Verified: the answer is the two by two zero matrix, and neither factor is zero. The chapter sets this up by first recalling the corresponding fact about ordinary numbers, so the contrast is explicit.
• The three surviving properties (§3.4.6, Part I p. 54). Regrouping; distribution from the left and from the right; and an identity matrix of matching order that leaves a square matrix alone. The chapter states all three without proof and says so — that sentence is honest.
• Example 16 (Part I pp. 54–55). A three by three, a three by two and a two by four, multiplied both ways round the bracket. Verified: both routes give the three by four matrix holding four, four, four and minus seven; thirty-five, minus two, minus thirty-nine and twenty-two; thirty-one, two, minus twenty-seven and eleven. Worth doing at least partly, because the two routes pass through completely different intermediate matrices — a three by two and a three by four one way, a three by four the other — and arriving at the same place is the whole point.
• Example 17 (Part I p. 56). Two three by three matrices and a three by one column; the sum times the column against the two products added. Verified: both come to the column holding ten, twenty and twenty-eight. The intermediate columns are nine, twelve and thirty, and one, eight and minus two.
• Example 18 (Part I pp. 56–57). A three by three holding one, two and three; three, minus two and one; four, two and one, with a cubic in it set equal to the zero matrix. Verified: the square is nineteen, four and eight; one, twelve and eight; fourteen, six and fifteen. The cube is sixty-three, forty-six and sixty-nine; sixty-nine, minus six and twenty-three; ninety-two, forty-six and sixty-three. Subtracting twenty-three times the matrix and forty times the identity leaves the zero matrix, position by position. Note the scalar attached to the identity — the constant term of a polynomial in a matrix has to be a matrix, and this is the first place that bites.
• Example 19 (Part I pp. 57–58). Campaign costs per contact in paise as a three by one column — forty, one hundred and fifty for telephone, house call and letter — against a two by three matrix of contact counts for two cities. Verified: the product is three hundred and forty thousand paise for the first city and seven hundred and twenty thousand for the second, that is three thousand four hundred rupees and seven thousand two hundred rupees. Note that the chapter forms the product with the count matrix on the left, which is the only order in which it is defined; use that to make section 12's point early.
• Exercise 3.2 Q13, Q14, Q15, Q16, Q17 and Q18 (Part I p. 60). Verified: Q13's rotation-like three by three satisfies the stated composition rule, by the Class XI compound-angle formulas. Q14 asks for two explicit disagreements, one two by two and one three by three, and both hold. Q15 evaluates to one, minus one and minus three; minus one, minus one and minus ten; minus five, four and four. Q16's cubic vanishes, checked position by position. Q17 gives the scalar as one. Q18 holds, and the cleanest route is to write the half-angle tangent in terms of the full angle's sine and cosine. Q17 is the item to work: four positions each give the same value of the unknown scalar, and a student who checks one and stops has not shown anything.
• Miscellaneous Example 23 (Part I p. 70). The nth power of a two by two rotation, proved by induction: base case, inductive hypothesis, and one step in which the compound-angle formulas collapse four entries. Verified: the step works exactly as printed. This is the only induction in the chapter and it is the best evidence that associativity is doing real work — the whole argument is a chain of products regrouped.
• Miscellaneous Example 25 (Part I pp. 71–72). Three two by two matrices, and an unknown fourth to be found from a product equation. Verified: the product of the two known factors is three and zero above forty-three and twenty-two, the four resulting scalar equations split into two independent pairs, and the unknown is minus one hundred and ninety-one and minus one hundred and ten above seventy-seven and forty-four. The chapter's first move is the one to teach: the order of the unknown is deduced from the equation before any entry is written down.
• Miscellaneous Exercise Q5, Q8, Q9 and Q11 (Part I pp. 72–73). Verified: Q5's quadratic in a two by two vanishes. Q8's unknown is one and minus two above two and zero, and it must be written on the left of the given factor, which is the point of the item. Q9's condition reduces to one less the first letter squared less the product of the other two, so the third option is right. Q11 expands using the fact that the matrix equals its own square, giving the identity, so the third option is right. Q11 is the best single item in the chapter for section 12, because expanding the bracket is only legal by the distributive law and the collapse only happens because powers behave.

Figures to have open

• A two-panel comparison of the products in Example 14, drawn to identical scale with the two disagreeing positions lit. The matrices are the chapter's; the lighting is added here. This is the load-bearing figure of the topic.
• A struck-through cancellation line for section 6, with Example 15's pair placed beside it as the refutation. Not in the book.
• A bracketing diagram for section 7 showing both routes through Example 16 and the two different intermediate matrices they pass through. Build the matrices with the repo's DataTable component.
• An induction loop for section 11 with four stations — base, hypothesis, step, next index. The repo's Cycle component carries this. The loop is the explanation's device; the chapter sets the argument out as running text.
• 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", the Remark on when both products exist, Part I p. 53
• Examples 13 and 14 and the run-in heading before them, Part I p. 53
• The Note on pairs that do commute, Part I p. 54
• Example 15 and the run-in heading before it, Part I p. 54
• §3.4.6, the three stated properties, Part I p. 54
• Examples 16, 17, 18 and 19, Part I pp. 54–58
• Exercise 3.2, questions 13 to 18, Part I p. 60
• Miscellaneous Examples 23 and 25, Part I pp. 70–72
• Miscellaneous Exercise on Chapter 3, questions 5, 8, 9 and 11, Part I pp. 72–73
• Summary, the associativity and distribution bullet, Part I p. 74