Orders one, two and three, and expanding along a chosen row

What to assume they know

• What a determinant is, and that only square arrays have one
• The order of a square array quoted as a single number
• Double-subscript notation, and the row index preceding the column index
• Powers of minus one, and the parity of a sum of two whole numbers
• Multiplying out brackets and collecting like terms in several letters
• The sine and cosine of an angle, and the identity relating their squares
• Solving a quadratic of the form an unknown squared equals a number

What they should be able to do

• Evaluate a determinant of order one, and say why the case is worth stating
• Evaluate a determinant of order two from its four entries, and describe the two tracks across the array that produce the two products
• Explain the reduction on which order three rests: each entry of a chosen line is paired with the smaller determinant left when its own row and column are struck out
• Attach the correct sign to each term from the two subscripts of its entry, and state the parity shortcut
• Expand a three-by-three determinant along a chosen row, and along a chosen column, and get the same value
• State how many distinct routes an order-three expansion has, and how many of them the chapter works through in print
• Choose the cheapest line to expand along by counting zeros first
• Predict what multiplying every entry of a square array by a fixed number does to its determinant, and name the exponent that appears
• Recognise a determinant that collapses to zero because of an identity among its entries rather than because of its zeros

Where it usually goes wrong

• "There is a formula for the three-by-three that I should memorise." There is a reduction, and the chapter is explicit about it. A student who memorises one row's expansion cannot use the zeros in a column, which is precisely the saving the chapter's own Example 3 is built to show.
• "You must expand along the first row." Any of six lines works and the chapter proves three of them agree before saying so. The freedom is the point; taking it away makes half the exercise set unnecessarily hard.
• "The sign alternates along the row, so it is always plus, minus, plus." It is plus, minus, plus along the top row, and minus, plus, minus along the second. The sign belongs to the position, not to the step number, and the chapter derives it from the subscript sum every single time for exactly this reason.
• "Deleting the row and column means deleting the entry." It means deleting the whole row and the whole column that the entry sits in — five entries gone in the order-three case, four survivors left. Drawing the strike-through fixes this in one shot.
• "Doubling a matrix doubles its determinant." It multiplies the determinant by two raised to the order. The chapter states this as a Remark and then makes two exercise items out of it, at orders two and three, so the exponent is clearly the thing being tested.
• "A determinant with a zero in it is zero." A zero entry kills one term of the expansion, not the value. Example 3 has two zeros and comes to minus fifty-two.
• "If the answer has no letters left in it, I have made a mistake." Example 2 and Miscellaneous Exercise Q1 both come out free of the variable they were written with. That is a result, not an error, and it is the first hint that a determinant can be an identity in disguise.
• "Six ways of expanding means six different answers to check." Six ways, one answer. The chapter verifies three of them and asserts the rest; an explanation should say which is which rather than implying all six are printed.

Questions to check understanding

• Evaluate a determinant of order two from four given entries, including a case where the answer is negative
• Evaluate a determinant of order two whose entries are algebraic and whose value is constant
• Expand a given three-by-three along a stated row, showing all three signs derived from subscripts rather than quoted
• Expand the same three-by-three along a stated column and confirm the two answers agree
• Given four three-by-three arrays, name the cheapest line to expand each along, and justify the choice by counting zeros
• Predict the determinant of a scaled array from the determinant of the original, at orders two and three — the form of Exercise 4.1 Q3 and Q4
• Set two determinants equal and solve for the unknown, reporting both roots where a square has been undone — the form of Exercise 4.1 Q7 and Q8
• Prove that a determinant containing an angle does not depend on it — the form of Miscellaneous Exercise Q1

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.

• Order one (§4.2.1, Part I p. 77). One line: a one-by-one array's determinant is its single entry. It looks like a joke and it is not — it is the base case that stops the reduction.
• Order two, with the book's own drawing (§4.2.2, Part I p. 77). The general two-by-two determinant is displayed inside bars with two dashed arrows drawn across it, one from the top-left entry to the bottom-right and one from the top-right to the bottom-left, crossing in the middle; the value is printed beside it as the first product minus the second. Read closely: both arrowheads point downward, so the drawing marks the two tracks but does not say which product is subtracted. If the explanation redraws it, add the sign cue and say that the book leaves it out.
• Examples 1 and 2 (§4.2.2, Part I p. 77). The numerical one has entries two and four on the top and minus one and two below; verified: two times two is four, four times minus one is minus four, and subtracting gives eight. The algebraic one has x and x plus one on top, x minus one and x below; verified: x times x is x squared, and the other product is x squared minus one, so the difference is one — every trace of x gone. Put them next to each other; the second is the first proof a student sees that a determinant can be constant.
• The reduction announced (§4.2.3, Part I p. 77). The chapter says an order-three determinant is settled by writing it in terms of order-two ones, gives that operation the name expansion, taken along whichever single line the reader picks, and states that there are six such routes and they all give the same value. Everything in this topic is the unpacking of that sentence.
• The four steps along the top row (§4.2.3, Part I p. 78). Step by step: take the first entry of the top row, attach the sign got from raising minus one to the sum of that entry's two subscripts, and multiply by the two-by-two determinant left when the entry's own row and column are struck out; repeat for the second and third entries; add the three terms. Read off the printed page: the top row is set in bold in the displayed array, so the line being expanded along is marked typographically. That device is used again on the same page family and is worth copying. Note: §4.2.3 never uses the words minor or cofactor. Both are defined two sections later, on Part I p. 84. The object arrives six pages before its name.
• The three sign exponents, and what the bracket beside them says (§4.2.3, Part I p. 78). Each step prints the exponent as a sum of the entry's two subscripts, with a bracketed gloss naming it the sum of the suffixes. Verified: the first and third terms get a plus and the middle term a minus, which is exactly the parity rule the chapter states as a Remark two pages later (Part I p. 80).
• The same array along the second row and along the first column (§4.2.3, Part I pp. 79–80). Both expansions are printed in full, both are multiplied out to six terms, and the chapter labels the three results so they can be compared. Read off the printed pages: the row or column being expanded along is set in bold each time — the second row in one display, the first column in the other. Verified: all three six-term results are the same sum, and the six terms are the six products that take one entry from each row and each column, three with a plus and three with a minus. That last sentence is an added organising remark, not the chapter's; flag it.
• Three routes worked, three handed over (§4.2.3, Part I p. 80). The chapter says outright that checking the remaining three routes — the third row and the second and third columns — is left to the reader, and then states the general conclusion anyway. The general claim in this book rests on three verified cases and an instruction.
• The three Remarks (§4.2.3, Part I p. 80). First, expand along whichever line carries the most zeros. Second, instead of computing a power of minus one, just read off whether the subscript sum is even or odd. Third, a worked pair of two-by-two arrays where one is twice the other, followed by the general statement that scaling a square array of order n by a number scales its determinant by that number raised to the power n. Verified on the chapter's own pair: the doubled array has value minus eight, the original minus two, and four times minus two is minus eight, with four being two squared and two being the order.
• Example 3 (Part I p. 80). A three-by-three with two zeros in the last column, expanded along that column; verified: only the entry four survives, its two-by-two determinant is minus one minus twelve, or minus thirteen, and four times minus thirteen is minus fifty-two. This is Remark (i) paying for itself in one line, and it is the best possible advertisement for counting zeros first.
• Example 4 (Part I pp. 80–81). A three-by-three built from sines and cosines of two angles, with zeros down its own diagonal, expanded along the top row. Verified: the first term dies on its zero entry, and the two survivors are equal in size and opposite in sign, so the value is zero. The interesting part is why — the array is antisymmetric, in that swapping the two subscripts of any entry flips its sign. That word is added here, not the chapter's, and the chapter offers no reason at all for the zero; it just computes it.
• Example 5 (Part I p. 81). Two determinants set equal, one containing an unknown twice; verified: the left side is three minus the unknown squared, the right is three minus eight, so the unknown squared is eight and the unknown is plus or minus two root two. Note the shape of the answer: two values, because a square was undone.
• Exercise 4.1 (Part I pp. 81–82). Eight items. Q1 and Q2 are direct evaluations, one numerical and two algebraic — verified: the numerical one is two times minus one minus four times minus five, which is minus two plus twenty, or eighteen; the trigonometric one is cosine squared plus sine squared, which is one; the polynomial one multiplies out to x squared minus x plus one times x plus one, minus x minus one times x plus one, which simplifies to x cubed plus one minus x squared plus one, that is x cubed minus x squared plus two. Q3 and Q4 are the scaling Remark applied at orders two and three, and they are the only two items in the chapter set with square brackets rather than bars, because they are about matrices being scaled. Q5 is four order-three evaluations. Q6 asks for the determinant of a bracketed array. Q7 and Q8 set two determinants equal and solve for an unknown, Q8 as a multiple choice. Verified for Q8: the equation reduces to the unknown squared equals thirty-six, so the unknown is plus or minus six, and the intended option is the one carrying both signs.
• Miscellaneous Exercise items that belong here (Part I pp. 99–100). Q1 asks for a proof that a certain three-by-three value does not depend on an angle; verified: expanding along the top row gives minus the unknown cubed, with every trigonometric term cancelling. Q2 is a three-by-three of products of sines and cosines of two angles; verified: expanding along the middle row, which carries a zero, gives one. Q5 and Q6 are evaluations in two letters; verified: Q5 comes to minus twice the sum of the two cubes, and Q6 comes to the product of the two letters. Q9 is a multiple choice about the range of a determinant as an angle runs over a full turn; verified: the value is two plus twice the square of the sine, so it runs over the closed interval from two to four, and the intended option is the closed one. See Notes for what Q5 and Q6 were designed to be solved with.

Figures to have open

• The general two-by-two inside bars with two tracks drawn across it. The chapter draws this on Part I p. 77 as two crossing dashed arrows; match the book's layout but add the sign cue it omits, and say that the cue is the explanation's addition.
• A strike-through movement for section 5: one entry chosen, its row and column faded, the four remaining entries sliding together into a two-by-two. The chapter describes the deletion in words on Part I p. 78 and draws nothing.
• A three-by-three grid of positions carrying subscript sums and signs, for section 6. Build it with the repo's DataTable component. The subscript sums are the chapter's own; the grid layout is added here.
• Three copies of one array with a different line highlighted in each, for section 9. The chapter itself sets the expanded line in bold on Part I pp. 78–79 — the top row in one display, the second row and then the first column in the others — so the highlight is the book's own device.
• No figure in this chapter is numbered, captioned or labelled Fig. Verified on the page image of every one of the twenty-eight pages. Every diagram the explanation draws here beyond the two arrow devices is its own.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 4 "Determinants", Part I pp. 76–103
• §4.2.1 Determinant of a matrix of order one, Part I p. 77
• §4.2.2 Determinant of a matrix of order two, with Examples 1 and 2, Part I p. 77
• §4.2.3 Determinant of a matrix of order 3 × 3, Steps 1 to 4 and the second-row and first-column expansions, Part I pp. 77–80
• The three Remarks following the expansions, Part I p. 80
• Examples 3, 4 and 5, Part I pp. 80–81
• Exercise 4.1, questions 1 to 8, Part I pp. 81–82
• Miscellaneous Exercise, questions 1, 2, 5, 6 and 9, Part I pp. 99–100
• Summary, the order-one, order-two and order-three bullets, Part I p. 101