Addition and scalar multiplication done entry by entry, and why orders must agree

What to assume they know

• The order of a matrix, and the fact that it is checked before entries are compared
• Equality of matrices, from the previous topic
• The double subscript, and what corresponding positions are
• Arithmetic with signed numbers, fractions, decimals and surds
• Basic trigonometric identities from Class XI, for two of the exercise items
• The zero matrix and its lack of an order restriction

What they should be able to do

• Add two matrices whose orders agree, working position by position
• State the precondition for a sum to exist, and say what happens when it fails
• Write the general definition of a sum in index form
• Multiply a matrix by a scalar, and state what the operation asks of the order
• Explain why a matrix's negative is one case of scaling rather than a separate idea
• Build a difference out of a sum and a scaling, and evaluate one
• Evaluate a combined expression such as twice one matrix minus another
• Work with entries that are expressions rather than numbers, including trigonometric ones
• Identify by inspection which of a set of matrix expressions are defined and which are not
• Recognise where the chapter uses vocabulary this edition no longer defines

Where it usually goes wrong

• "If the orders differ, add what you can and leave the rest." Nothing is produced at all. The Note on Part I p. 44 exists to say so, and a student who pads the shorter matrix with zeros has invented an operation the chapter does not have.
• "Scaling needs the orders to match too." It needs nothing. One number and one matrix of any order whatever, and every entry is multiplied. Half the content of this topic is that the two operations differ here.
• "Multiplying a matrix by three multiplies one row, or the first entry." Every entry. The chapter's intermediate display on Part I p. 45, with the six products written out unevaluated, is there to prevent exactly this.
• "The negative of a matrix flips the sign of the numbers but not the letters." It scales by minus one and every entry goes, including an entry that is a letter. The chapter's own instance on Part I p. 46 has a letter in it for this reason.
• "Subtraction is its own operation with its own rules." It is a sum with a scaling wrapped round the second matrix, which is why it inherits the same order precondition and nothing more needs proving about it.
• "Twice A minus B means take A minus B and double it." The scaling binds to the matrix it is written against. Example 7 is worth stepping through slowly for this reason alone.
• "An entry that is an expression has to be left alone." Exercise 3.2 Q2 has three items whose entries collapse — two by algebra and one by a Class XI identity — and the collapse is the point of the item.
• "Adding matrices is a new kind of operation, unlike anything from earlier chapters." The chapter's own Note calls it an instance of a general notion, using a term this edition no longer defines. Section 11 handles this honestly; see Notes.

Questions to check understanding

• Add two matrices whose orders agree, and state the order of the result
• Say whether a stated sum is defined, and justify the verdict from the orders alone
• Multiply a matrix by a scalar, including a negative and a fractional one
• Write down the negative of a stated matrix, including one with a letter entry
• Evaluate an expression combining a scaling with a difference — the form of Example 7 and of Exercise 3.2 Q1(iii)
• Simplify a sum whose entries are algebraic expressions — the form of Exercise 3.2 Q2(ii)
• Simplify a sum whose entries collapse under a Class XI trigonometric identity — the form of Exercise 3.2 Q2(iv) and Q6
• Verify that regrouping three matrices does not change a sum — the form of Exercise 3.2 Q4

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 two-factory build-up (§3.4.1, Part I p. 43). Two three by two matrices, rows labelled one, two and three for price categories and columns labelled boys and girls. Read this off the page image: the text layer drops both matrices entirely. The first holds eighty and sixty, seventy-five and sixty-five, ninety and eighty-five; the second holds ninety and fifty, seventy and fifty-five, seventy-five and seventy-five. The chapter then writes the six totals out as unevaluated sums, which is the right thing to show: the answer's structure is visible before any arithmetic happens.
• The general definition (Part I p. 44). Stated first on a two by three pair with lettered entries, then in index form for any shared order, the sum's entry at each position being the sum of the two entries there. The chapter attaches the order to the result explicitly.
• Example 6 (Part I p. 44). Two matrices of order two by three. The first holds root three, one and minus one above two, three and zero; the second holds two, root five and one above minus two, three and one half. Verified: the sum is two plus root three, one plus root five and zero above zero, six and one half. The text layer loses both radicals — the second matrix's middle entry reads as a plain five in extraction and is root five on the page. Read the entries off the image. Note also that the chapter writes each position's sum with the second matrix's entry first; harmless, and worth not copying.
• The Note refusing a sum (Part I p. 44). Item 1 gives a two by two and a two by three and states that no sum exists for them. Verified: the second matrix has a third column with nothing to pair against. Show the two matrices and leave the unpaired column visibly unmatched — that picture is the entire argument.
• The doubled factory (§3.4.2, Part I p. 45). The first factory's three by two matrix reprinted, then the same six positions written as two times each entry, then evaluated to one hundred and sixty and one hundred and twenty, one hundred and fifty and one hundred and thirty, one hundred and eighty and one hundred and seventy. Read off the page image; the text layer drops the first of the three displays. The intermediate display, with the multiplications unresolved, is the one.
• The general scaling rule and its instance (Part I p. 45). Stated in index form, then instanced on a three by three matrix holding three, one and one point five; root five, seven and minus three; two, zero and five, tripled to nine, three and four point five; three root five, twenty-one and minus nine; six, zero and fifteen. Verified position by position. Root five again survives only on the image.
• The negative (Part I pp. 45–46). Defined as scaling by minus one, then instanced on a two by two holding three and one above minus five and x, giving minus three and minus one above five and minus x. Verified: the letter entry is negated like any other, which is the point of including it.
• The difference (Part I p. 46). Defined in index form and then immediately rewritten as a sum with the negative, so that subtraction inherits everything proved about addition rather than needing its own account.
• Example 7 (Part I p. 46). Twice a two by three matrix holding one, two and three above two, three and one, less a two by three holding three, minus one and three above minus one, zero and two. Verified: twice the first is two, four and six above four, six and two; the negative of the second is minus three, one and minus three above one, zero and minus two; the sum of those is minus one, five and three above five, six and zero. The chapter does the subtraction in exactly that order — scale, negate, add.
• Exercise 3.2 Q1, parts (i), (ii) and (iii) (Part I p. 58). Three two by two matrices are given: the first holds two and four above three and two, the second one and three above minus two and five, the third minus two and five above three and four. Verified: their sum in the first two is three and seven above one and seven; the first minus the second is one and one above five and minus three; and three times the first less the third is eight and seven above six and two. Parts (iv) and (v) are products and belong to the multiplication topics.
• Exercise 3.2 Q2 (Part I p. 58). Four sums whose entries are expressions. Verified: (i) gives a plus a and b plus b above b minus b and a plus a, that is twice a and twice b above zero and twice a. (ii) collapses to a perfect square in each position — the top row gives the square of the sum of a and b and the square of the sum of b and c, and the bottom row the square of the difference of a and c and the square of the difference of a and b. (iii) is plain arithmetic on nine positions. (iv) uses only the Class XI identity that the two squares add to one, so every one of its four positions becomes one, and the answer is the two by two array of ones. Part (iv) is the best item in the exercise and the chapter gives no worked instance of an entry that simplifies by identity.
• Exercise 3.2 Q4, Q5 and Q6 (Part I p. 59). Q4 asks for two sums and a verification that regrouping does not change the answer; Q5 combines two scalings and a difference on two three by three matrices of fractions; Q6 asks for a sum of two scaled two by two matrices of trigonometric entries. Verified for Q6: the four positions come to the square of the cosine plus the square of the sine on the diagonal and zero off it, so the answer is the two by two identity. That result is worth showing because it is the first identity matrix in the chapter to arrive as the answer to something rather than as a definition.

Figures to have open

• The two factory matrices from Part I p. 43 with their printed row numbers and their boys and girls column labels, drawn as a pair. Build them with the repo's DataTable component. They are the chapter's own and carry sections 1 and 2.
• An order-mismatch panel for section 4: a two by two and a two by three set side by side with the third column of the second left visibly unpartnered. The matrices are the chapter's from Part I p. 44; the unpartnered rendering is added here.
• The unevaluated scaling display from Part I p. 45 — six positions each showing two times a number, before the products are worked out. This is the chapter's own middle step and it is easy to skip; do not.
• A three-stage strip for section 9 showing scale, negate and add as separate frames.
• 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 opening, Part I p. 43
• §3.4.1, the two-factory build-up and the general definition, Part I pp. 43–44
• Example 6 and the Note refusing a sum, Part I p. 44
• §3.4.2, scaling, the negative and the difference, Part I pp. 45–46
• Example 7, Part I p. 46
• Exercise 3.2, questions 1 to 6, Part I pp. 58–59
• Summary, the scaling, negative and difference bullets, Part I p. 74