Packing three equations into AX = B and reading the solution off the inverse

What to assume they know

• The inverse of a square array, and the formula that produces it from the adjoint and the determinant
• That an inverse exists exactly when the determinant is not zero
• Multiplying a matrix by a column, row into column
• That matrix multiplication does not generally commute, from Chapter 3
• The identity matrix, and that multiplying by it changes nothing
• That multiplication of matrices may be regrouped without changing the product
• Solving a pair of linear equations by elimination, from an earlier class

What they should be able to do

• Rewrite a system of two or three linear equations as one matrix equation, and name the three arrays involved
• Check that carrying out the multiplication reproduces the original equations line by line
• State the condition under which the method applies, and test it before doing anything else
• Carry out the four-line rearrangement that isolates the column of unknowns
• Say which side the inverse must be applied on, and what goes wrong on the other side
• Explain why the solution obtained this way is the only one
• Solve a system in two unknowns and a system in three unknowns by this method
• Turn a worded problem into three equations and then into one matrix equation
• Use a product that has been handed to you, rather than computing an inverse
• Recognise a system that becomes linear after a substitution, and solve it

Where it usually goes wrong

• "Divide both sides by the coefficient array." There is no division for matrices. There is multiplication by an inverse, and it has a side. Say the word "inverse" every time; the moment an explanation says "divide", the side stops mattering to the student.
• "It does not matter which side you multiply on." It does. Matrix multiplication does not commute and the chapter names the side explicitly at the one step where it matters. Multiplying on the wrong side leaves an array stranded between the inverse and the column, and nothing cancels.
• "Check the determinant at the end." Check it first. If it is zero the whole method is unavailable and everything computed afterwards is wasted. The condition is the first line of Case I for exactly this reason.
• "The column on the right holds the answers." It holds the right-hand sides of the equations. The answers appear only after the multiplication by the inverse. Mixing these two columns up is the commonest setup error.
• "The unknowns have to be written in the order they appear in the first equation." They have to be written in the same order in every equation, and a missing unknown contributes a zero coefficient. Exercise 4.5 Q11 has an equation with only two unknowns in it and the zero must be written in.
• "A worded problem gives you the equations directly." One of the three conditions in Example 18 is a comparison and has to be rearranged before it is an equation at all. That rearrangement is where marks are lost.
• "You always have to compute the inverse." Not when you are handed a product that turns out to be the identity. The chapter's own Miscellaneous Example does exactly that and computes no cofactor at all.
• "Fractional answers mean I made a mistake." Two of the four two-unknown items and one of the four three-unknown items in Exercise 4.5 have fractional answers. They are correct.

Questions to check understanding

• Rewrite a given system as a matrix equation and name the three arrays
• Test whether the method applies to a given system, before solving it
• Solve a system in two unknowns by this method — the form of Exercise 4.5 Q7 to Q10
• Solve a system in three unknowns by this method — the form of Exercise 4.5 Q11 to Q14
• Find an inverse first and then use it on a system — the form of Exercise 4.5 Q15
• Turn a worded problem into three equations and solve it — the form of Exercise 4.5 Q16
• Solve a system whose unknowns appear underneath, by substituting — the form of Miscellaneous Exercise Q7
• Given two arrays whose product is the identity, solve a system without computing any cofactor

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 framing (§4.6, Part I p. 93). One sentence saying that determinants and matrices are about to be applied to solving systems in two or three unknowns and to testing them. Open on it; it is the first time the chapter says what all the machinery was for.
• The packing (§4.6.1, Part I p. 94). Three general equations in three unknowns are set out, then three arrays are named — the square array of coefficients, the column of unknowns, and the column of right-hand sides — and the system is rewritten as one product equalling one column. Read off the printed page: all three sit inside square brackets. There is not a single determinant bar in the packing; the bars reappear only when the condition is tested. That switch is the visual cue for what kind of object is in play. Verified, and the chapter does not do it: multiplying the packed form out reproduces the three printed equations exactly, row by row. Do that check once. Students who have not seen it treat the packing as notation rather than as a true statement.
• Case I, the condition (§4.6.1, Part I p. 94). If the coefficient array is on the non-zero side of the line, its inverse exists. Everything downstream depends on this one test and it costs one determinant.
• The four lines (§4.6.1, Part I p. 94). Multiply both sides on the left by the inverse; regroup the two arrays on the left, which the chapter attributes to the regrouping property by name; the inner pair becomes the identity; the identity leaves the column of unknowns alone. The chapter labels the first step with the side it is applied on — that label is the single most important word on the page and an explanation that drops it is teaching the wrong habit.
• Where uniqueness comes from (§4.6.1, Part I p. 94). The chapter's own reason is that an inverse is unique, so the column it produces is unique too. Say it; students who have only ever met "one solution" as a geometric statement about crossing lines benefit from seeing it fall out of algebra instead.
• Example 16 (Part I p. 95). Two equations in two unknowns. Verified: the coefficient array has entries two and five on top, three and two below, its determinant is four minus fifteen, or minus eleven, so the method applies; the inverse is the adjoint over minus eleven; and the answer is three and minus one. Note that this is the pair the module's opening video was told to use as its concrete motivating example, chosen there precisely so that the loop could be closed here; the chapter's own opening on Part I p. 76 is in letters, not numbers, and points at nothing.
• Example 17 (Part I pp. 95–96). Three equations in three unknowns. Verified: the determinant is minus seventeen; the nine cofactors are minus one, minus eight, minus ten; minus five, minus six, one; minus one, nine, seven; and the answer is one, two and three. A clean full-length worked example and the one to build section 8 on.
• Example 18 (Part I pp. 96–97). A worded problem: three numbers with three stated conditions, one of which has to be rearranged before it is an equation at all. Verified: the determinant is nine, the adjoint has rows seven, minus three, two; three, zero, minus three; minus one, three, one; and the numbers are one, two and three. The translation step is the teaching content here — the third condition is stated as a comparison and must be moved into standard form before the packing can happen. This example contains a dropped minus sign in its printed coefficient array. See Notes. The rest of the worked solution is internally consistent and correct; only the displayed array is wrong.
• Miscellaneous Example 19 (Part I pp. 98–99). A product of two given arrays is checked to be the identity, which identifies the second as the first's inverse without any cofactor being computed, and the system is then solved with it. Verified: the product is the identity, and the solution is zero, five and three. This is the best item in the module for showing that the method is about having an inverse, not about the adjoint recipe. The two arrays in this example's problem statement are printed without any of their minus signs, while the same two arrays in the solution one line below carry them. See Notes; this is the most serious printed defect in the chapter.
• Exercise 4.5 Q7 to Q10 (Part I p. 97). Four systems in two unknowns. Verified determinants and answers: one, giving two and minus three; eleven, giving minus five elevenths and twelve elevenths; minus eleven, giving minus six elevenths and minus nineteen elevenths; and four, giving minus one and four. Two of the four have fractional answers, which is worth flagging — a student who expects whole numbers will assume an error.
• Exercise 4.5 Q11 to Q14 (Part I p. 97). Four systems in three unknowns. Verified determinants and answers: thirty-four, giving one, a half and minus three halves; ten, giving two, minus one and one; forty, giving one, two and minus one; and four, giving two, one and three. Q11 is the one with a fractional right-hand side and a zero coefficient, so it is the item that breaks a student who has been pattern-matching.
• Exercise 4.5 Q15 (Part I p. 98). An inverse is asked for first and then used on a system. Verified: the determinant is minus one, the inverse has rows zero, one, minus two; minus two, nine, minus twenty-three; minus one, five, minus thirteen; and the answer is one, two and three. The two-part phrasing is the point — it separates building the tool from using it.
• Exercise 4.5 Q16 (Part I p. 98). A shopping problem in three commodities and three totals. Verified: the determinant is fifty, and the three prices per unit weight are five, eight and eight. The arithmetic is heavier than anywhere else in the exercise and the answer is clean, which is a good sign the item was built backwards from it.
• Miscellaneous Exercise Q7 (Part I p. 100). Three equations whose unknowns appear only underneath. Verified: replacing each reciprocal by a new unknown makes the system linear with determinant twelve hundred, the three new unknowns come out as a half, a third and a fifth, and the original three are therefore two, three and five. The chapter gives no hint that a substitution is wanted, and no worked example anywhere in the chapter uses one. Flag that.
• The Summary bullets (Part I p. 102). Three belong here: the packing, the solution formula with its condition attached, and the consistency line. Read off the page image: the Summary states the condition alongside the formula, which is more careful than the body's own Case I heading manages.

Figures to have open

• A side-by-side of three stacked equations and the packed matrix sentence, for section 1. The chapter prints both on Part I p. 94 and draws no connection between them; the figure from one to the other is added here.
• A row-into-column multiplication trace for section 3, each row of the coefficient array sweeping across the column of unknowns. Not in the book; the chapter states the packing and never verifies it.
• A table of exercise items against determinant and answer shape for section 11. Build it with the repo's DataTable component. The determinants and answers are added derivations.
• 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 Applications of Determinants and Matrices, the opening sentence, Part I p. 93
• §4.6.1, the packing and Case I, Part I p. 94
• Example 16, Part I p. 95; Example 17, Part I pp. 95–96; Example 18, Part I pp. 96–97
• Exercise 4.5, questions 7 to 16, Part I pp. 97–98
• Miscellaneous Example 19, Part I pp. 98–99
• Miscellaneous Exercise, question 7, Part I p. 100
• Summary, the packing, solution and consistency bullets, Part I p. 102