Deleting a row and a column: minors, and the sign that turns one into a cofactor

What to assume they know

• Expanding a determinant of order three along a chosen row or column
• Powers of minus one, and how the parity of a sum controls the sign
• Double-subscript notation, with the row index read before the column index
• Determinants of order two, computed from four entries
• The order of a square array, quoted as a single number
• That an entry sits in exactly one row and exactly one column

What they should be able to do

• State what the minor of an entry is, and produce it by deleting the entry's own row and its own column
• Say what the order of a minor is, given the order of the determinant it came from
• Compute a named minor of a numbered three-by-three array
• State what a cofactor is, and produce one from the corresponding minor and the entry's two subscripts
• Say for which positions the minor and the cofactor coincide, and for which they differ by a sign
• Write out all four minors and all four cofactors of a two-by-two determinant
• Write two general cofactors of a lettered three-by-three array in full
• Write the order-three expansion in its compact form as a sum of three products
• State that any of the six lines gives the same value in that compact form

Where it usually goes wrong

• "The minor is what is left when you delete the entry." You delete the entry's whole row and its whole column — five entries in the order-three case, four survivors. Deleting just the entry leaves eight entries and no square array at all.
• "Minor and cofactor are two words for the same thing." They agree at four of the nine positions in an order-three array and disagree in sign at the other five. The whole reason both names exist is that one of them carries the sign and one does not.
• "The sign depends on the value of the entry." It depends only on where the entry sits. Exercise 4.3 Q1 has a zero whose cofactor sign is undetectable, and Q2's identity array has six of them; use one of those to make the point that the sign was determined before anybody looked at the number.
• "The sign alternates along a row, so I can just count plus, minus, plus." That works along the top row and inverts on the second. The reliable rule is the parity of the two subscripts added, which the chapter uses every single time rather than quoting an alternation.
• "A minor of a three-by-three is a number, so it has no order." It is a determinant of order two that happens to evaluate to a number. Keeping it a determinant is what makes the reduction recursive, and it is why the chapter states the order rule as a Remark of its own.
• "Every entry has one minor, so a three-by-three has three." It has nine — one per entry. Exercise 4.3 Q2 asks for all eighteen numbers across two arrays precisely to break this.
• "The compact form is a different formula from the long expansion." It is the same expansion with two names substituted in. Show the substitution happening rather than presenting the short form as new.
• "Because six lines all work, I should check more than one." Six lines, one value. Checking a second is a way to catch an arithmetic slip, not a requirement of the definition.

Questions to check understanding

• Produce a named minor of a given three-by-three array, showing the deletion
• Produce the corresponding cofactor and state which subscript sum decided its sign
• State the order of a minor taken from a determinant of stated order
• Write out all four minors and all four cofactors of a two-by-two — the form of Exercise 4.3 Q1
• Write out all nine minors and all nine cofactors of a three-by-three — the form of Exercise 4.3 Q2
• Identify, for a given array, the positions at which minor and cofactor agree
• Rewrite a fully expanded order-three determinant in compact form, naming each substitution as it is made

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 section's own statement of purpose (§4.4, Part I p. 84). One sentence: the point of what follows is to write the expansion compactly, using two named objects. Read it as a promise and then keep the explanation honest about it — every beat of this topic either defines one of the two objects or spends it.
• Definition 1, the minor (§4.4, Part I p. 84). The minor of an entry is the determinant you are left with after deleting that entry's row and that entry's column, and it is written with a capital M carrying the entry's two subscripts. Note that the definition is phrased against a determinant, not against a matrix — read off the printed page, the object being cut up sits inside vertical bars in both Example 8 and Example 9. That is consistent through the whole section and worth preserving.
• The Remark on order (§4.4, Part I p. 84). A minor taken from a determinant of order n, with n at least two, has order one less. The lower bound is doing real work: at order one there is nothing left after the deletion, which is why the chapter refuses to define the case. Verified consequence, supplied by the explanation: at order two every minor is a single entry, so its value is that entry — which is exactly why the two-by-two formula looks like two products rather than like an expansion. The chapter never joins those two dots.
• Example 8 (§4.4, Part I p. 84). The array is the nine counting numbers in order, and the entry six is located by its row, which is the second, and its column, which is the third; its minor is the determinant of the four survivors. Verified: those survivors are one and two on top and seven and eight below, giving eight minus fourteen, which is minus six. Note the coincidence — the entry is six and its minor is minus six — and say out loud that it is a coincidence, because a fair number of students will read it as a rule.
• Definition 2, the cofactor (§4.4, Part I p. 84). The cofactor of an entry is minus one raised to the sum of that entry's two subscripts, multiplied by that entry's minor, and it is written with a capital A carrying the same two subscripts. So the cofactor is the minor when the subscript sum is even and its negative when the sum is odd, and nothing else ever happens.
• Example 9 (§4.4, Part I p. 84). All four minors and all four cofactors of a two-by-two whose entries are one and minus two on top, four and three below. Verified: the four minors are three, four, minus two and one, taken in the order first row then second row; the four cofactors are three, minus four, two and one. Exactly the two off-diagonal positions flip, which is the cleanest possible demonstration of Definition 2 and the reason the chapter puts this example immediately after it.
• Example 10 (§4.5 preamble, Part I p. 85). Two cofactors of a general lettered three-by-three, written out in full: the one at the top-left, whose subscript sum is even so the sign changes nothing, and the one directly below it, whose subscript sum is odd so every term flips. Setting the two side by side is the whole lesson of section 8 — the minors are different objects, and on top of that one of them has been negated.
• The Remark that spends the definitions (Part I p. 85). Expanding the general three-by-three along its top row and then substituting the cofactor names collapses the display to three products added together. The chapter then states the general form: the value is the sum you get by pairing each entry of any one line with its own cofactor. This is the payoff and it should be the climax of the explanation, not a footnote. This Remark carries a printed slip — it refers the reader to an example number that does not exist in this chapter. See Notes.
• Six lines, one statement (Part I p. 85). The chapter says the same calculation can be run along the other two rows and all three columns. That is the compact restatement of what §4.2.3 established the long way.
• Exercise 4.3 Q1 (Part I p. 87). Two two-by-two determinants, one numerical and one lettered, each wanting all four minors and all four cofactors. Verified for the numerical one, whose entries are two and minus four on top and zero and three below: the minors are three, zero, minus four and two, and the cofactors are three, zero, four and two. Note the zero — its sign flip is invisible, which is a good moment to point out that the sign is attached to the position and not to the value. Verified for the lettered one: the minors are the four entries taken across the diagonals, and the two off-diagonal cofactors carry a minus.
• Exercise 4.3 Q2 (Part I p. 87). Two three-by-three determinants, eighteen minors and eighteen cofactors between them. Verified for the identity array: the three cofactors on the main diagonal are one and the other six are zero, so every sign flip in that item is invisible. Verified for the second array, whose rows are one, zero, four; three, five, minus one; and zero, one, two: the nine minors are eleven, six, three; minus four, two, one; minus twenty, minus thirteen, five — and the cofactors flip exactly the four positions whose subscript sums are odd, giving eleven, minus six, three; four, two, minus one; minus twenty, thirteen, five.
• The Summary bullets (Part I p. 101). The Summary restates both definitions in one line each. Read off the page image: the minor bullet survives intact; the cofactor bullet prints a dropped word and reads as though it were missing its verb. Confirmed. Do not show the printed line.

Figures to have open

• A strike-through device: one entry chosen, its row and column faded, the four remaining entries sliding together. The chapter describes the deletion in words on Part I p. 84 and draws nothing, so this is added here and it is the single most useful picture in the topic.
• A three-by-three grid of positions carrying subscript sums, parities and signs, for section 6. Build it with the repo's DataTable component. The subscript sums are the chapter's own; the grid is added here.
• A two-column layout of minors against cofactors for section 7. The four values are the chapter's own from Example 9; the side-by-side layout is added here.
• No figure in this chapter is numbered, captioned or labelled Fig. Verified on the page image of every one of the twenty-eight pages. The only two pieces of line art in the whole chapter are the crossing arrows inside the two-by-two determinant on Part I p. 77 and the looped arrows over the two-by-two matrix on Part I p. 88; neither belongs to this topic.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 4 "Determinants", Part I pp. 76–103
• §4.4 Minors and Cofactors, the opening sentence, Definition 1 and the Remark on order, Part I p. 84
• Example 8 and Definition 2, Part I p. 84; Example 9, Part I p. 84
• Example 10 and the Remark that follows it, Part I p. 85
• Exercise 4.3, questions 1 and 2, Part I p. 87
• Summary, the minor and cofactor bullets, Part I p. 101