Consistent or inconsistent: what a vanishing determinant does and does not decide

What to assume they know

• Packing a system of equations into one matrix equation
• The determinant of the coefficient array, and what a non-zero value licenses
• The adjoint of a square array, and how to build it
• Multiplying a square array by a column
• The zero matrix, and what it means for a column to be it
• Two straight lines that cross once, never, or lie on top of each other
• That a system may have no answer at all

What they should be able to do

• State the two outcomes a system can be sorted into, before computing anything
• Compute the determinant of a coefficient array and read off which branch of the decision the system falls into
• Say what is lost, and what is not lost, when that determinant is zero
• Carry out the second test by multiplying the adjoint into the right-hand column
• Conclude no solution when that product is not the zero column, and justify the conclusion
• Say why the other outcome of that test leaves two possibilities open, and name both
• Sort a two-unknown system into its branch, and check the answer against the geometry of two lines
• Sort a three-unknown system into its branch, including one whose coefficients contain a letter
• Read the Summary's three-line verdict as a decision procedure and apply it cold

Where it usually goes wrong

• "A zero determinant means there is no solution." It means there is not exactly one. There may be none and there may be infinitely many, and the whole point of the second test is that the determinant alone cannot say which.
• "The second test decides everything." It decides one branch and explicitly fails on the other. The chapter says so in one sentence and an explanation that turns the test into a two-way rule is contradicting the book it is teaching.
• "Consistent means solvable by this method." Consistent means an answer exists. A system with infinitely many answers is consistent and this method will not produce them.
• "Inconsistent means the equations are wrong." It means they contradict one another. Each is a perfectly good equation; together they demand the impossible.
• "If the determinant is zero, compute the inverse anyway and see." There is no inverse to compute. The adjoint still exists, which is exactly why the second test is phrased in terms of the adjoint and not the inverse.
• "Two equations that look different must be independent." The second equation of Exercise 4.5 Q3 is twice the first on its left side and not on its right. Looking different is not being independent.
• "A letter in a coefficient is just another number." It is a case split. Exercise 4.5 Q4 changes its answer depending on whether that letter is zero, and the printed item does not warn you.
• "The chapter promised to stay with uniquely solvable systems, so this cannot come up." It printed that promise and then broke it on the next page and in its own exercise. See Notes.

Questions to check understanding

• Sort a given system into one of the two outcomes, stating which test was used
• Compute a coefficient determinant and say what it does and does not settle
• Carry out the second test on a system with a zero determinant and state the verdict — the form of Exercise 4.5 Q3 and Q5
• Say what remains undecided when the second test returns the zero column, and name both remaining possibilities
• Sort a three-unknown system whose coefficients contain a parameter, giving both cases — the form of Exercise 4.5 Q4
• Produce two systems with the same zero determinant, one consistent and one not
• Apply the Summary's three-branch verdict to a system given cold

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 definitions (§4.6, Part I p. 94). Two short bold-headed statements, one for each outcome, sorting systems by whether any answer exists. Read off the printed page: neither carries a definition number, unlike the five numbered definitions earlier in the chapter, and the key word in each is set in italic inside the sentence. So they look like passing remarks and are in fact the vocabulary the whole section runs on.
• The boxed restriction, and it does not survive contact with the next two pages (§4.6, Part I p. 94). A tinted Note directly beneath those two definitions announces that the chapter will keep to systems with exactly one answer. Read off the printed page: the Note sits immediately above the subsection heading, and the very next page prints Case II, which is entirely about arrays with no inverse, ending in a sentence that names both no answer and infinitely many. Exercise 4.5 then opens with six items asking the reader to sort systems, and verified: three of the six are not uniquely solvable. The Note is therefore contradicted by the two pages that follow it and by the exercise it leads into. See Notes.
• Case II (§4.6.1, Part I p. 94). If the determinant is zero, the inverse is unavailable, and the chapter's instruction is to compute a different product instead: the adjoint of the coefficient array multiplied into the column of right-hand sides.
• The first branch of the second test (§4.6.1, Part I p. 94). If that product is not the zero column, no answer exists. Verified, and worth showing because the chapter gives no reason at all: suppose a solution existed. Multiplying the packed equation on the left by the adjoint turns the left side into the determinant times the column of unknowns, which is the zero column when the determinant is zero — so the right side must be the zero column too. A non-zero result therefore rules a solution out. Three lines, entirely from Theorem 1, and it converts a rule into an argument.
• The second branch (§4.6.1, Part I p. 95). If the product is the zero column, the chapter says the system may fall either way, according to whether it has infinitely many answers or none. That sentence is doing something unusual for a textbook — it is admitting that the test has run out.
• Why the test runs out (not in the book). The chapter offers none. Supply one: a zero determinant means the equations are not independent of one another, and that alone cannot tell you whether the redundant equation agrees with the others or contradicts them. Two lines lying on top of each other and two parallel lines both give a zero determinant, and they have opposite answers. Flag this as the explanation's account.
• Exercise 4.5 Q1 and Q2 (Part I p. 97). Two systems in two unknowns. Verified: the first has determinant minus one and the second has determinant three, so both are on the branch where exactly one answer exists. Two easy items whose purpose is to establish that the determinant is the first thing computed.
• Exercise 4.5 Q3, and it is the one to build a section on (Part I p. 97). Two equations in two unknowns whose determinant is zero. Verified: the determinant is six minus six, or zero; the adjoint has entries six and minus three on top, minus two and one below; and pushing the right-hand column through it gives six and minus two, which is not the zero column. So there is no answer. Verified independently, and this is the beat that lands: the left side of the second equation is exactly twice the left side of the first, while its right side is not twice the first's — the two equations demand different things of the same quantity. Draw the two parallel lines.
• Exercise 4.5 Q5 (Part I p. 97). Three equations in three unknowns, one of which is missing an unknown. Verified: the determinant is zero; the adjoint has rows minus five, ten, five; minus three, six, three; minus six, twelve, six; and the product with the right-hand column is minus five, minus three, minus six, which is not the zero column. So there is no answer. This is the only three-unknown worked instance of the first branch available anywhere in the chapter, and the chapter does not work it.
• Exercise 4.5 Q6 (Part I p. 97). Three equations in three unknowns. Verified: the determinant is fifty-one, so exactly one answer exists and no second test is needed.
• Exercise 4.5 Q4, and it needs a caveat the printed item does not carry (Part I p. 97). Three equations, the third of which has a letter as its coefficient throughout. Verified: the determinant of the coefficient array is that letter itself. So the system has exactly one answer whenever the letter is not zero, and when it is zero the third equation reads as a false numerical statement and there is no answer at all — which the chapter's own second test confirms, since the product with the right-hand column then comes out as minus four, zero, four. The printed item places no condition on the letter, so the honest answer is conditional. Say so; an explanation that reports a single verdict here is teaching a student to ignore a parameter.
• The Summary's verdict (Part I p. 102). Three sub-bullets: a non-zero determinant gives exactly one answer; a zero determinant with a non-zero adjoint product gives none; a zero determinant with a zero adjoint product leaves the matter open. Read off the page image: this is the only place in the chapter where the three branches appear together as one list, and it is a better presentation than the body's own. Build section 10 on it.

Figures to have open

• Two lines crossing, two lines lying on top of each other, and two parallel lines, drawn to the same scale, for sections 7 and 8. The chapter draws no lines anywhere and never mentions the geometry; this is added here and it is what makes the undecidable branch feel inevitable rather than like a textbook giving up.
• A decision tree with three leaves for section 10. Build it with the repo's Network component. The three branches are the chapter's own, from Part I p. 102; drawing them as a tree is added here.
• A side-by-side of the printed restriction and the two later pages that ignore it, for section 2. Quote nothing; paraphrase the restriction and point at the section heading and the exercise instruction that contradict it.
• 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 line art in the chapter is on Part I pp. 77 and 88, and neither belongs to this topic.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 4 "Determinants", Part I pp. 76–103
• §4.6, the two unnumbered definitions and the boxed Note, Part I p. 94
• §4.6.1, Case II and its two branches, Part I pp. 94–95
• Exercise 4.5, questions 1 to 6, Part I p. 97
• Summary, the consistency bullet and the three-branch bullet, Part I p. 102