Singular against non-singular, and the one test an inverse has to pass

What to assume they know

• The adjoint of a square array, and the theorem relating an array, its adjoint and its determinant
• The inverse of a matrix from Chapter 3, and that it is unique when it exists
• Multiplying a matrix by a number, and by the identity
• That matrix multiplication does not generally commute, from Chapter 3
• Multiplying two square arrays of the same order
• Rearranging an equation by multiplying both sides on the same side
• Squaring a matrix, and reading a polynomial equation satisfied by one

What they should be able to do

• State the two definitions this section introduces and place any given array on one side of the line by one computation
• State the theorem that ties invertibility to the determinant, in both directions
• Follow the easy half of its proof: from an inverse existing to the determinant being non-zero
• Follow the hard half: from the determinant being non-zero to an inverse, by dividing the adjoint theorem through
• Write the resulting formula for an inverse and say exactly which step required the determinant to be non-zero
• Compute the inverse of a given two-by-two and a given three-by-three by that formula
• Decide, before computing anything else, whether a given array has an inverse at all
• Use the product rule for determinants, and state why the inverse of a product reverses the order
• Extract an inverse from a polynomial equation that the array satisfies, without computing any cofactor

Where it usually goes wrong

• "Singular means the array has no adjoint." It has one. The adjoint never divides by anything. What a singular array lacks is an inverse, and the adjoint theorem still holds for it with zero on the right.
• "If the determinant is zero I should look for the inverse more carefully." There is nothing to find. Theorem 4 is an exactly-when statement, so a zero determinant closes the question. Exercise 4.4's instruction to say when an inverse does not exist is testing whether the student stops.
• "The inverse is the adjoint." Only when the determinant is one, which is exactly the situation in the chapter's own Example 13 — so the example that teaches the formula is the one example that hides it. Pair it with a determinant that is not one.
• "The inverse of a product is the product of the inverses." In the reverse order. The chapter verifies the reversal twice, in Example 14 and in Exercise 4.4 Q12, and never explains it; an explanation should, because the reason is one line and the rule is otherwise pure memorisation.
• "Theorem 4 was proved from the definition of an inverse." Its hard half was proved from the adjoint theorem, which was itself only verified at order three, and its easy half used the product rule for determinants, which is stated without proof. Say what rests on what.
• "Post multiplying and pre multiplying are the same thing." They are not, and the chapter is careful about it — Example 15 names the side explicitly, and the next section names the other side. Getting it wrong is the standard way this argument fails.
• "You always need cofactors to find an inverse." Not when the array satisfies a polynomial equation you have been handed. Four exercise items and one worked example use that route and never compute a cofactor.
• "An array cannot be its own inverse." One item in the exercise set is. It is a good thirty seconds and the chapter passes over it in silence.

Questions to check understanding

• Decide whether a given array has an inverse, by one computation, and name the case
• Compute the inverse of a given two-by-two by the adjoint formula
• Compute the inverse of a given three-by-three by the adjoint formula — the form of Exercise 4.4 Q7 to Q11
• State which single step of the existence proof requires the determinant to be non-zero
• Verify the reversal rule on a given pair of arrays — the form of Exercise 4.4 Q12
• Extract an inverse from a polynomial equation the array satisfies — the form of Exercise 4.4 Q13, Q15 and Q16
• Find the coefficients that make such an equation hold — the form of Exercise 4.4 Q14
• Give the determinant of an inverse in terms of the original — the form of Exercise 4.4 Q18

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.

• Definitions 4 and 5, and the examples attached to them (§4.5.1, Part I p. 89). One name for arrays whose determinant is zero and another for the rest, each illustrated immediately. Verified: the first example's entries are one and two on top, four and eight below, and its determinant is eight minus eight, which is zero; the second's are one and two on top, three and four below, and its determinant is four minus six, which is minus two. The pair is well chosen — the two arrays differ only in the bottom row and land on opposite sides of the line. Note: the chapter spells the second name both with and without a hyphen, sometimes three lines apart. Read on the page images of Part I pp. 89 and 102. Pick one spelling and keep it.
• Theorem 2 (§4.5.1, Part I p. 89). A product of two arrays that are both on the non-zero side, taken in either order, stays on that side. Stated without proof.
• Theorem 3 (§4.5.1, Part I p. 89). The determinant of a product is the product of the two determinants. Also stated without proof, and it is the one the rest of the section actually leans on — it is used in the theorem below and again in the exponent rule on the previous page. Say plainly that it is being taken on trust. Verified on the chapter's own Example 14 arrays: one has determinant minus eleven and the other has determinant one, and their product has determinant minus eleven. The chapter computes all three numbers and never puts them together as a check on Theorem 3.
• Theorem 4, and it is the point of the section (§4.5.1, Part I p. 90). An array has an inverse exactly when its determinant is not zero. Unlike the three above it, this one is proved, and the proof is short enough to run on air.
• The easy half of the proof (Part I p. 90). Suppose an inverse exists. Then the two multiply to the identity; take determinants of both sides; by Theorem 3 the left becomes a product of two numbers and the right is one; a product of two numbers equal to one cannot have a zero factor. Three lines, no machinery beyond Theorem 3.
• The hard half (Part I p. 90). Suppose the determinant is not zero. Then the adjoint theorem may be divided through by it, and what stands on the left is the original array multiplied by a named array, in either order, giving the identity — which is precisely the definition of an inverse. This is where the whole module is spent. Every cofactor computed since Part I p. 84 exists to make that one division available.
• The formula (Part I p. 90). The inverse is the adjoint divided by the determinant. One line.
• Example 13, second half (Part I pp. 90–91). The same three-by-three whose adjoint was built in the previous topic, now inverted. Verified: its determinant is one, so the inverse and the adjoint coincide — which is a pedagogical hazard rather than a convenience, because the division is invisible. Pair it deliberately with an example whose determinant is not one.
• Example 14 (Part I p. 91). Two two-by-twos, their product inverted, and the claim that the inverse of the product is the product of the inverses in the reverse order. Verified: the first array's determinant is minus eleven, the second's is one, the product's is minus eleven, and multiplying the two inverses in reversed order reproduces the product's inverse entry for entry. The reversal is the interesting part and the chapter does not explain it.
• Example 15 (§4.5.1, Part I p. 92). A two-by-two satisfying a quadratic matrix equation, from which the inverse is extracted without a single cofactor being computed. Verified: the array squared has entries seven and twelve on top, four and seven below; subtracting four times the array and adding the identity gives the zero matrix; and rearranging gives the inverse as four times the identity minus the array, which is two and minus three on top, minus one and two below. Cross-checked against the adjoint formula: the determinant is one and the adjoint is that same array. This is the section's best trick and it is worth its own beat.
• Exercise 4.4 Q5 to Q11 (Part I pp. 92–93). Seven inverses, two at order two and five at order three, with the instruction to say when one does not exist. Verified determinants: fourteen, thirteen, ten, minus three, minus three, minus one, and minus one. All seven exist — no item in this block is singular, which is worth knowing before the explanation promises a counterexample from the exercise set. Two of the seven are worth pulling out. The trigonometric one at Q11 comes out equal to its own inverse, which the chapter does not remark on and which is the most memorable single fact in the exercise. And the array at Q10 is the same array the chapter inverts by a completely different route in its Miscellaneous Example on Part I p. 98; setting the two side by side is free and the book never does it.
• Exercise 4.4 Q12 (Part I p. 93). The reversal rule verified on a numerical pair. Verified: the two determinants are one and minus two, and the product's determinant is minus two, consistent with Theorem 3.
• Exercise 4.4 Q13 to Q16 (Part I p. 93). Four items in the style of Example 15, each giving a polynomial equation and asking for the inverse from it. Verified: Q13's array has determinant seven and inverse the adjoint over seven; Q14 asks instead for the two coefficients that make the equation hold and they are minus four and one; Q15's array has determinant minus eleven and Q16's has determinant four, and in both cases rearranging the given cubic gives the same inverse as the adjoint formula does. Q14 is the odd one out and an explanation should flag it, because it runs the trick backwards.
• Exercise 4.4 Q17 and Q18 (Part I p. 93). Two multiple choices. The first is the exponent rule from the previous topic. Verified for Q18: an array and its inverse have determinants whose product is one, so the determinant of the inverse is the reciprocal, and the intended option is the reciprocal one.
• Miscellaneous Exercise Q3, Q4 and Q8 (Part I pp. 99–100). Q3 gives one array's inverse and a second array outright and asks for the inverse of the product; verified: the second array's determinant is one, so its inverse is its own adjoint, and multiplying in the reversed order gives rows nine, minus three, five; minus two, one, zero; one, zero, two. Q4's second half asks for a double inverse returning the original. Q8 is a multiple choice on the inverse of an array with entries only on its diagonal; verified: the inverse has the three reciprocals in the same places, and the intended option is the one showing them with no factor in front.
• The Summary bullets (Part I p. 102). Four of them belong here: the two names as a single either-or line, the definition of an inverse through the identity, the theorem in one clause, and the formula. Read off the page image: the Summary states the theorem correctly and carries no proof and no hint of one, which is the right choice for a summary and worth pointing out to a student who revises only from it.

Figures to have open

• A number line with zero marked, arrays dropping onto one side or the other as their determinants are computed, for sections 1 and 2. The chapter draws nothing here; this is added here and it is what makes a definition feel like a decision.
• A cancellation movement for section 9: two arrays and two inverses in a row, the inner pair meeting first and collapsing to the identity. Not in the book; the chapter verifies the rule numerically and draws nothing.
• A table of the exercise items against route and determinant for section 11. Build it with the repo's DataTable component. The determinants are added derivations; the table 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 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
• §4.5.1, Definitions 4 and 5 with their examples, Part I p. 89
• Theorems 2 and 3, Part I p. 89
• Theorem 4 and its proof, and the inverse formula, Part I p. 90
• Example 13's second half, Part I pp. 90–91; Example 14, Part I p. 91; Example 15, Part I p. 92
• Exercise 4.4, questions 5 to 18, Part I pp. 92–93
• Miscellaneous Exercise, questions 3, 4 and 8, Part I pp. 99–100
• Summary, the singular, inverse, existence and formula bullets, Part I p. 102