The laws addition and scaling obey, and what they buy you

What to assume they know

• Addition, scaling, the negative and the difference of matrices, from the previous topic
• Equality of matrices, and the fact that one matrix equation is many scalar equations
• The zero matrix, and the fact that it exists at every order
• Commutativity, associativity and distributivity for ordinary numbers
• Solving a linear equation by doing the same thing to both sides
• Solving a pair of simultaneous equations by elimination
• Percentages of a quantity, for the applied example

What they should be able to do

• State each of the four laws of matrix addition and each of the two laws of scaling
• Reproduce the three-step proof pattern: descend to a position, use the fact about numbers, ascend again
• Say which ordinary-number fact each matrix law rests on
• Explain what role the zero matrix plays and what role the negative plays, and why both are needed to solve for an unknown matrix
• Solve a matrix equation for an unknown matrix, naming the law used at each step
• Recover two unknown matrices from two equations relating them
• Recover unknown scalars from an equation combining scalings and sums
• Read an applied problem stated in matrices and answer sums, differences and percentage questions from it
• Spot a printed derivation whose written form does not match the argument it is making

Where it usually goes wrong

• "These laws are obvious, so the proofs are a formality." The proofs are where the student learns that a matrix statement is settled at one arbitrary position. That move is used again for the transpose laws and for the symmetry theorems later in the chapter, and a student who skipped it here has to invent it there.
• "The laws hold for matrices of any orders." Every one of the addition laws carries the same order condition the sum itself carries, and the Summary repeats the condition on each line. A law about an operation is only as wide as the operation.
• "The zero matrix is a single object." There is one at every order, and the identity law is about the one whose order matches. The chapter's phrasing attaches the order deliberately.
• "Subtracting a matrix from both sides is allowed because subtraction is allowed." It is allowed because the negative exists and cancels, and because addition regroups so that the cancellation can be brought together. Example 8 spells out both steps and a student who compresses them cannot say why the compressed step works.
• "Scaling and adding are two unrelated operations that happen to appear in the same section." The two laws of §3.4.4 are exactly the statements that they interact predictably, and without them nothing in Example 8 after the first line is legal.
• "A printed derivation is always correct as printed." The second derivation in §3.4.4 is not; see the Notes. Say the corrected line and move on — do not show the printed one and do not make a lesson out of the error.
• "A zero in a difference matrix means the entry was missing." In Example 11 it means the two months were equal in that variety. Zero is a value.
• "Two matrix equations in two unknown matrices need a new technique." They need the same elimination as two scalar equations, licensed by these six laws. Exercise 3.2 Q7(ii) is the item that makes this visible.

Questions to check understanding

• State a named law of matrix addition and give its order condition
• Prove a named law by descending to one position and using the number fact there
• Name the law licensing a stated step in a matrix manipulation
• Solve for an unknown matrix in an equation combining scalings and sums — the form of Example 8 and of Exercise 3.2 Q8
• Recover two unknown matrices from two equations relating them — the form of Example 9 and of Exercise 3.2 Q7
• Recover unknown scalars from an equation of scaled matrices — the form of Example 10 and of Exercise 3.2 Q9, Q10 and Q11
• Handle an equality with the unknown on both sides — the form of Exercise 3.2 Q12
• Answer combined, difference and percentage questions from a pair of data matrices — the form of Example 11

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 four laws of addition (§3.4.3, Part I pp. 46–47). Numbered (i) to (iv), each with a bold run-in name. The first two carry a printed derivation; the second two are stated and instanced rather than derived, because there is nothing to derive once the zero matrix and the negative have been defined. The associativity derivation carries the chapter's own parenthetical question at the step where the number fact is used — that question mark is an invitation: the regrouping is legitimate because ordinary addition regroups.
• The two laws of scaling (§3.4.4, Part I p. 47). Stated first as a pair on one line, then derived. The printed block has two defects, both confirmed on a 300 dot per inch the printed page of the source. First, the two statements are labelled (i) and (ii), and the two derivations below them are labelled (ii) and (iii), so the list runs (i), (ii), (ii), (iii) and no derivation carries the label (i). Second, the derivation of the second law prints a plus sign where an equals sign belongs: as set, the line reads as a sum of three bracketed matrices, where the argument requires the first bracket to equal the sum of the other two. Verified: with the equals sign restored the line is correct and three-step like all the others. Section 7 should show the corrected line and should not show the printed one.
• Example 8 (Part I pp. 47–48). Twice a three by two matrix plus three times an unknown equals five times another three by two matrix. The first matrix holds eight and zero, four and minus two, three and six; the second holds two and minus two, four and two, minus five and one. The chapter's solution is the centre of this topic: it subtracts, reorders, cancels, drops the zero matrix and finally scales by a third, annotating four of those five steps with the law that permits it. Verified: five times the second matrix is ten and minus ten, twenty and ten, minus twenty-five and five; twice the first is sixteen and zero, eight and minus four, six and twelve; their difference is minus six and minus ten, twelve and fourteen, minus thirty-one and minus seven; and a third of that is minus two and minus ten thirds, four and fourteen thirds, minus thirty-one thirds and minus seven thirds. Show the annotations beside the algebra — they are the reason this example is here rather than in the previous topic.
• Example 9 (Part I p. 48). Two unknown matrices given by their sum, five and two above zero and nine, and their difference, three and six above zero and minus one. Verified: adding the two equations gives twice the first unknown as eight and eight above zero and eight, so the first is four and four above zero and four; subtracting gives twice the second as two and minus four above zero and ten, so the second is one and minus two above zero and five. The chapter writes the cancellation explicitly as a pairing that vanishes, which is worth keeping — it is the additive inverse law doing visible work.
• Example 10 (Part I pp. 48–49). Twice a two by two whose entries include two unknowns, plus a two by two of numbers, equal to a two by two of numbers. Verified: the scaling is applied first, then the sum, then the equality is read position by position, giving twice the first unknown plus three equal to seven and twice the second minus four equal to fourteen, so the unknowns are two and nine. This is the topic's cleanest link back to the equality definition: the laws build the left side, and equality dismantles it.
• Example 11 (Part I pp. 49–50). Two farmers, three varieties of rice, sales in rupees for two months as two two by three matrices with named rows and named columns. Read both matrices off the page image; the text layer drops them. September holds ten thousand, twenty thousand and thirty thousand for the first farmer and fifty thousand, thirty thousand and ten thousand for the second. October holds five thousand, ten thousand and six thousand, then twenty thousand, ten thousand and ten thousand. Three questions: combined sales, the drop from one month to the next, and two per cent of the second month. Verified: the combined matrix is fifteen thousand, thirty thousand and thirty-six thousand above seventy thousand, forty thousand and twenty thousand; the difference is five thousand, ten thousand and twenty-four thousand above thirty thousand, twenty thousand and zero; and two per cent of October is one hundred, two hundred and one hundred and twenty above four hundred, two hundred and two hundred. The zero in the difference matrix is worth a sentence — one variety sold identically in both months, and a zero entry in a difference is information, not an absence.
• Exercise 3.2 Q7 (Part I p. 59). Two items of the same shape as Example 9. Verified, (i): the two unknowns are five and zero above one and four, and two and zero above one and one. (ii) is harder because the coefficients are two and three rather than one and one; verified by elimination, the first unknown is two fifths and minus twelve fifths above minus eleven fifths and three, and the second is two fifths and thirteen fifths above fourteen fifths and minus two. Item (ii) is the best evidence in the exercise that these laws are what make elimination legal on matrices at all.
• Exercise 3.2 Q8, Q9, Q10, Q11 and Q12 (Part I p. 59). Verified: Q8 gives the unknown as minus one and minus one above minus two and minus one. Q9 gives the two unknowns as three and three. Q10 gives three, six, nine and six for the four letters, read in the order the question names them. Q11 gives three and minus four. Q12 gives two, four, one and three. Q12 is the only one of the five in which an unknown appears on both sides of the equality, and it should be worked for that reason.
• The Summary bullets (Part I p. 74). Six lines restating these laws, each with its order condition attached. Worth showing at the close. Note that the Summary states the laws and prints none of the derivations, which is appropriate for a summary and is worth saying so the student does not think something is missing.

Figures to have open

• A one-position zoom for section 2: a bracketed sum shrinking onto a single highlighted entry and expanding again. This is an added device and it carries the whole proof pattern; reuse it in section 6.
• An annotated ladder for section 8, Example 8's five lines down the left and the licensing law printed against each. The lines and the annotations are both the chapter's; the pairing is the explanation's arrangement of them.
• The two sales matrices from Part I p. 49 with their printed row and column labels, and the three derived matrices beneath them. Build them with the repo's DataTable component.
• A corrected rendering of the second scaling derivation for section 7. Do not reproduce the printed line.
• 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", §3.4.3, the four laws of addition, Part I pp. 46–47
• §3.4.4, the two laws of scaling, Part I p. 47
• Examples 8, 9 and 10, Part I pp. 47–49
• Example 11, the two-farmer sales problem, Part I pp. 49–50
• Exercise 3.2, questions 7 to 12, Part I p. 59
• Summary, the six law bullets, Part I p. 74