Why cofactors borrowed from the wrong row always sum to zero

What to assume they know

• Minors and cofactors, and how a cofactor's sign comes from two subscripts
• Expanding a determinant along a chosen row or column
• The compact form of an expansion as entries paired with their own cofactors
• Determinants of order two, computed quickly
• That a determinant can be zero without any of its entries being zero
• Adding and subtracting signed products carefully

What they should be able to do

• State the correct pairing that produces a determinant, and contrast it with the mismatched pairing
• Predict the value of a mismatched sum before computing it, and be right every time
• Follow the chapter's argument: substitute the cofactor definitions, recognise the result as an expansion, and identify the array it expands
• Name the fact that closes the argument, and say plainly that this book does not state it
• Verify a mismatched sum numerically on a three-by-three whose determinant is not zero, so the contrast is visible
• Evaluate a determinant using the cofactors of a named row or a named column
• Distinguish, among four written sums, the one that evaluates to the determinant from the ones that evaluate to zero
• Say what this result is being saved up for, two sections ahead

Where it usually goes wrong

• "It is zero because the array was chosen conveniently." It is zero for every square array, including ones whose own determinant is large. Show the chapter's Example 11 array with both numbers visible — its determinant is minus twenty-eight and its mismatched sum is zero — and the suspicion dies.
• "So a sum of entries times cofactors is always zero." Only when the entries and the cofactors come from different lines. Matched, the same shape of sum gives the determinant. The whole content of the result is the word different.
• "The cofactors change when you borrow them." They do not. A cofactor belongs to a position in the original array and is computed once. Borrowing means using it in a sum it was not built for, not recomputing it.
• "The chapter proved that a determinant with two matching rows is zero." It did not. It used the fact in one parenthesis and states it nowhere. This is the single most important thing to be honest about in this topic — see Notes.
• "It works for rows, so somebody should check columns separately." The chapter says the same argument runs for columns and does not repeat it. The explanation may say the same, but should say that it is an assertion rather than a second proof.
• "This is a curiosity with no use." It is exactly half of the reason the adjoint works, two sections later. Without it, the product of an array with its adjoint would have no reason to be diagonal. Plant the forward reference.
• "Zero here means the array is singular." No. The array in Example 11 is not singular at all. The zero is a property of the mismatch, not of the array.
• "The signs in the printed argument are the ones I should copy." They are not — see Notes. Run the substitution on air with signs derived from subscripts, as the chapter itself does everywhere else.

Questions to check understanding

• Predict, without computing, the value of a stated mismatched sum, and justify the prediction
• Compute a mismatched sum on a given three-by-three and confirm it is zero
• Compute the same array's determinant and state why the two answers differ
• Evaluate a determinant using the cofactors of a named row — the form of Exercise 4.3 Q3
• Evaluate a lettered determinant using the cofactors of a named column, and factorise the answer — the form of Exercise 4.3 Q4
• Choose, from four written sums, the one equal to the determinant, and say what each of the others equals — the form of Exercise 4.3 Q5
• State the fact about matching rows that the chapter's argument needs, and give a one-line proof of it at order two

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 correct pairing, restated (Part I p. 85). Immediately above the Note, the chapter has just concluded that pairing every entry of any single line with its own cofactor and adding gives the determinant. Open on that, because the Note is only interesting against it.
• The Note's claim (the boxed Note, §4.4, Part I p. 85). Take the entries of one line and the cofactors belonging to a different line, multiply correspondingly, add, and the total is zero. Read off the printed page: the Note sits in a tinted box with a pointing-hand icon, the same device the chapter uses twice more, and the box is the only place in the whole chapter where this result appears outside the Summary.
• The three-line argument (Part I p. 85). The chapter takes the entries of the top row against the cofactors of the second row, substitutes each cofactor as a signed minor, and observes that the resulting three-term expression is the expansion of a particular three-by-three array. That array has the top row copied into the second row position, and the chapter closes by saying the two match, so the value is zero. Two printed defects live inside those three lines. Both were confirmed and both are recorded in Notes below.
• The array the argument lands on (Part I p. 85). Read off the printed page: its first two rows are the same three letters in the same order, and its third row is the original array's third row. Draw it; the whole argument is visible in that one picture and invisible in the algebra.
• The word that closes it, and the gap behind it (Part I p. 85). The parenthetical justification is that the two rows match. That is an appeal to a general fact — a determinant with two matching rows is zero — and this book states that fact nowhere. The word the chapter uses for the matching occurs exactly once in the entire chapter, in this parenthesis; checked against the extracted text of all twenty-eight folios. The explanation has two honest options: prove the small case, or say plainly that the fact is being imported from outside this book. It must not imply the chapter established it. Verified, and cheap enough to show: an order-two determinant with its two rows the same is one product minus the identical product, so it is zero in one line. Doing that much makes the import feel small rather than hand-waved.
• Example 11 (§4.4, Part I p. 86). A three-by-three with entries two, minus three, five; six, zero, four; one, five, minus seven. All nine minors and all nine cofactors are computed and then the entries of the top row are paired with the cofactors of the bottom row. Verified: the nine cofactors are minus twenty, forty-six, thirty; four, minus nineteen, minus thirteen; minus twelve, twenty-two, eighteen. The mismatched sum is two times minus twelve, plus minus three times twenty-two, plus five times eighteen — that is minus twenty-four, minus sixty-six, plus ninety, which is zero. Verified and worth adding, because the chapter does not: the same array's own determinant, from the correct pairing, is minus twenty-eight. Show the two numbers together. A student who sees only the zero half-suspects the array was rigged; seeing minus twenty-eight beside it kills that.
• Exercise 4.3 Q3 (Part I p. 87). Evaluate a numbered three-by-three using the cofactors of its second row. Verified: those three cofactors are seven, seven and minus seven, the second-row entries are two, zero and one, and the determinant comes to seven. Note what the item is really drilling — the freedom to pick a line, cashed through the compact form.
• Exercise 4.3 Q4 (Part I p. 87). Evaluate a lettered three-by-three using the cofactors of its third column. Verified: the three cofactors are the three differences of the letters taken in rotation, and the determinant multiplies out to the product of the three differences — the first letter minus the second, times the second minus the third, times the third minus the first. Checked numerically as well as symbolically, on two separate substitutions. This is the hardest item in the exercise and the only one whose answer is a factorisation rather than a number.
• Exercise 4.3 Q5 (Part I p. 87). A multiple choice offering four sums built from entries and cofactors, asking which one is the determinant. Verified: the intended option is the one that pairs the entries of the first column with the cofactors of the first column; the option pairing the first row's entries with the third row's cofactors is zero by this topic's own result, and so is the option pairing the second row's entries with the first row's cofactors. The remaining distractor mixes indices in a way that matches neither pattern. This item is the reason the topic exists as a separate video: a student who has only met the correct pairing cannot eliminate three of the four options.
• The Summary bullet (Part I p. 101). The Summary carries this result as its last line, with the same mismatched sum written out. What it does not carry is the argument, or the fact about matching rows that the argument needs. A student revising from the Summary gets an assertion with no support at all.

Figures to have open

• Two copies of one array with pairing arrows drawn on them — matched in the first, crossing between two rows in the second. The chapter draws no arrows here; this is added here and it is the picture the topic turns on.
• The array with a duplicated row, drawn with both copies highlighted in the same colour, for section 5. The chapter prints that array on Part I p. 85 and marks nothing; the highlight is added here.
• A three-by-three grid holding the nine cofactors of the Example 11 array, for section 7. Build it with the repo's DataTable component. The nine values are the chapter's own; the grid 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 tinted Note box on Part I p. 85 carries a small pointing-hand icon and no other artwork.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 4 "Determinants", Part I pp. 76–103
• §4.4, the concluding statement about the correct pairing, Part I p. 85
• The boxed Note and its three-line argument, §4.4, Part I p. 85
• Example 11, Part I p. 86
• Exercise 4.3, questions 3, 4 and 5, Part I p. 87
• Summary, the mismatched-sum bullet, Part I p. 101