What it takes to be an inverse, and why there can only ever be one

What to assume they know

• Matrix multiplication, the condition for a product to exist, and the order of the answer
• The identity matrix, and the fact that it leaves a square matrix unchanged
• Square matrices, and the fact that only they can be multiplied by themselves
• The failure of commutativity, and the reason a two-sided condition is not automatic
• The associative law for a triple product
• The transpose reversal law, as the model for the pattern this section repeats
• Equality of matrices

What they should be able to do

• State Definition 6, naming both products it demands
• Explain why a matrix with unequal counts cannot have an inverse at all
• Verify that a given pair are inverses by computing both products
• State that the relation runs both ways, and say what that rules out
• State the uniqueness result and reproduce its proof
• Identify the step of that proof where associativity is used, and the two steps where the two halves of the definition are used
• State the reversal law for the inverse of a product and apply it
• Reproduce the reversal law's proof, naming what is multiplied on which side at each step
• Choose the correct characterisation of an inverse pair from a set of options
• Say what this section supplies and what it does not, and where the missing piece is found

Where it usually goes wrong

• "One product equal to the identity is enough." The definition demands both. For square matrices over the reals one does in fact imply the other, but nothing in this chapter proves that, and the uniqueness proof on Part I p. 69 uses both halves. Treat the demand as printed.
• "The inverse is like a reciprocal, so it exists for anything that is not zero." Nothing in this chapter says which square matrices have inverses. It gives the definition and two theorems about matrices already known to be invertible, and stops.
• "A rectangular matrix might have a one-sided inverse." Not a question this chapter asks. The Note rules out an inverse in the sense defined, and the reason is an order argument, not an arithmetic one.
• "The inverse of a product is the product of the inverses in the same order." It is the reversed product, exactly as with the transpose. Teach the two reversal laws together; the second one costs almost nothing once the first has landed.
• "Uniqueness is obvious." It needs associativity and both halves of the definition, and it is a five-link chain. Run every link.
• "You can divide by a matrix." No division is defined in this chapter, and the earlier failure of cancellation is the reason. Every manipulation with an inverse is a multiplication on a stated side.
• "The chapter will now show me how to compute one." It will not. Section 10 should say so plainly, say what an exam will actually ask on this material, and point forward honestly; see Notes for what may and may not be claimed.
• "An identity matrix and the number one are interchangeable." They are not, and the chapter's own proof prints the digit where the matrix belongs. Use the matrix.

Questions to check understanding

• State the condition for two matrices to be inverses of each other, and choose it from a set of options — the form of Exercise 3.4 Q1
• Verify that a given pair are inverses by computing both products
• Explain why a matrix whose two counts differ cannot have an inverse
• State and prove that a square matrix has at most one inverse
• Name the fact used at each link of that proof
• State the reversal law for the inverse of a product
• Reproduce the reversal law's proof, saying on which side each multiplication happens
• Given that two matrices are invertible, write down the inverse of their product without computing anything

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.

• Definition 6 (§3.7, Part I p. 68). Fixes the order at the outset, requires a partner of that same order, and then writes the demand as a single chain: one product, the other product, and the identity, all equal. It closes by naming the notation and the adjective. The chain is the whole definition and section 2 should read it left to right and then right to left, because the chain is symmetric and the definition inherits that.
• The worked pair (Part I p. 68). A two by two holding two and three above one and two, and a two by two holding two and minus three above minus one and two. Verified: the first product is four minus three and minus six plus six above two minus two and minus three plus four, which is the identity; and the second product is the identity too. The chapter prints the first product's intermediate entries and reports the second product's result without intermediates. Show both sets of intermediates — the point of the example is that two computations are needed, and a page that shows one and asserts the other undercuts it.
• The Note (Part I p. 69). Two items. The first rules out an inverse for a matrix whose counts differ, arguing from the requirement that both products exist and be equal. The second states that the relation runs both ways. Verified, item 1: if the first matrix has order m by n with m and n different, then for both products to exist the second must be n by m; the two products then have orders m by m and n by n, which cannot be equal to each other, let alone both equal to one identity. That is the argument the Note compresses into one sentence and section 3 should run it out.
• Theorem 3 and its proof (Part I p. 69). The claim is that a square matrix cannot have two different inverses. The proof supposes two, labels the two defining chains, and then runs a five-link equality: the first candidate, then the first candidate times the identity, then the first candidate times the product of the matrix with the second candidate, then a regrouping, then the identity times the second candidate, then the second candidate. Three different facts are used in that chain. — the identity property, associativity, and one half of each candidate's defining condition. Verified: the regrouping is exactly the associative law of §3.4.6, and the two halves used are the second candidate's left-hand product and the first candidate's right-hand product, which is why a one-sided definition would not have been enough.
• Theorem 4 and its proof (Part I p. 69). The claim is that the inverse of a product turns the factors around, exactly as the transpose does. The proof is a ladder: start from the defining condition for the product, multiply on the left by the first matrix's inverse, collapse, multiply on the left by the second matrix's inverse, collapse again. Verified step by step. Say out loud at each rung which side the multiplication happens on — the whole proof works because every multiplication is on the same side, and a student who multiplies on the right at any rung gets stuck.
• A printed slip in that proof (Part I p. 69). The proof's opening line sets the product of a matrix with its inverse equal to the digit one, where the identity matrix is required. Confirmed on a 300 dot per inch the printed page of the source. Every subsequent line of the same proof uses the identity correctly, so the intent is not in doubt. Do not show the printed line; write the identity.
• Exercise 3.4 (Part I p. 69). One question, four options, and nothing else. Verified: the condition is that both products equal the identity, so the fourth option is right; the first option is the definition of commuting and not of inverting, the second and third both put a zero matrix where an identity belongs. The whole exercise is this one item — checked on the printed page image and confirmed on a 300 dot per inch the printed page, on which roughly the bottom third of the page is blank. Its emptiness is itself a finding and section 10 should be built on it; see Notes.
• The Summary (Part I p. 74). Two bullets: the definition, and the uniqueness. Verified against the page images of Part I pp. 73 and 74: the reversal law of Theorem 4 is not in the Summary. A student revising only from the Summary leaves this section with a definition and a uniqueness claim and no reversal law at all. Put Theorem 4 in the explanation's closing card for that reason.

Figures to have open

• A both-directions chain for section 2: the defining equality drawn once and traversable each way, so the symmetry of the relation is visible before section 5 asserts it. An added staging of the chapter's own line.
• An order-tracking strip for section 3: a rectangular candidate's counts carried through both products until the two answers visibly have different orders. An added argument, expanding a sentence the Note compresses.
• A side-by-side of the two products of the worked pair from Part I p. 68, both with intermediates shown. The second matrix's intermediates are added here, since the page reports only its result.
• A five-link annotated chain for section 7 and a rung-by-rung ladder for section 9, with the side of each multiplication marked on the ladder.
• No redraw of a textbook figure is possible or needed: this chapter prints no figure at all. Confirmed on the page image of every one of the forty-two pages.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 3 "Matrices", §3.7, Definition 6 and the worked pair, Part I p. 68
• The Note, items 1 and 2, Part I p. 69
• Theorem 3 and its proof, Part I p. 69
• Theorem 4 and its proof, Part I p. 69
• Exercise 3.4, its single question, Part I p. 69
• Summary, the inverse definition and the uniqueness bullets, Part I p. 74