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

0:00Numbers have reciprocals. Multiply by three, then by a third, and you are back where you started. Matrices multiply, so it is fair to ask the same question. Is there a matrix that undoes another one? The answer is a definition, and the definition asks for more than you might expect. It does not ask for one product to come out right. It asks for two. That second demand looks like tidiness. It is not.

0:32It is what forces both matrices to be square, it is what makes the relation run both ways, and it is what makes the uniqueness proof work at all. So this video is about a definition, and about what each half of it is carrying. Here is the definition. Take a square matrix A of order n. If there is another square matrix B of the same order n with A times B equal to B times A equal to the identity, then B is called an inverse of A.

1:07Read that middle line slowly, because it is a chain of three equalities and every link matters. A times B is the identity. B times A is the identity. And therefore the two products are equal to each other. Now read the chain from the right instead. B times A is the identity, A times B is the identity. That is the same sentence with A and B exchanged. The chain is symmetric, so the definition inherits the symmetry.

1:41Neither of the two matrices is the special one. They are partners, and the word inverse describes the pair. One more thing the definition fixes before it says anything else: both matrices are square, and both have the same order. Hold on to that. We will come back and see that it was not a restriction somebody chose. It is a consequence. The definition tells you how to CHECK a candidate. It does not tell you how to find one.

2:14So here is a matrix and a candidate handed over with it. The matrix holds two and three above one and two. The candidate holds two and minus three above minus one and two. Multiply the first by the second. Top left: four and minus three, which is one. Top right: minus six and six, which is zero. Bottom left: two and minus two, zero. Bottom right: minus three and four, one.

2:45That is the identity. One product down. Now the other order, and this is the half that usually gets skipped. Top left: four and minus three again, one. Top right: six and minus six, zero. Bottom left: minus two and two, zero. Bottom right: minus three and four, one. The identity again. Both products, both checked, and only now is the candidate confirmed. Two multiplications. That is the whole procedure this definition offers, and doing one of them is doing half the job.

3:25A reasonable objection: if the first product came out right, surely the second one had to. For square matrices over ordinary numbers that happens to be true. But nothing here proves it, and the reason it is dangerous is that it is only true for square ones. Take all two by twos built from minus one, zero and one, and pair each of them with every candidate of the same kind. Six thousand five hundred and sixty-one ordered pairs.

3:59Count the pairs where exactly one of the two products is the identity and the other is not. Zero. Over square matrices the second demand really is free. Now do the same thing with matrices that are not square. Two rows and three columns, paired with three rows and two columns. Four thousand and ninety-six pairs. Count the pairs where exactly one product is the identity. Forty-two. The second demand is not free there at all.

4:32And the number of pairs where BOTH products are the identity, among those four thousand and ninety-six, is zero. So the definition's second half is doing something, and where it does something, it does it decisively. Why can a matrix whose two counts differ not have an inverse? The argument is about orders, and no entry is looked at anywhere in it. List every order from one by one up to four by four. Sixteen of them.

5:05Pair each with each, and you have two hundred and fifty-six ordered pairs of orders. For both products to exist at all, the second matrix's counts have to be the first one's counts swapped over. That leaves sixteen pairs out of two hundred and fifty-six. Now look at what those sixteen produce. If the first matrix is m by n, one product is m by m and the other is n by n.

5:34For those two answers even to be comparable, m and n have to agree. Four of the sixteen. The other twelve give two answers of different orders, and two matrices of different orders are not equal. They are not unequal either. The question cannot be put. Concretely: a matrix with two rows and three columns can only be paired with one that has three rows and two columns. The two products are then two by two and three by three, and no single identity is both.

6:11So it is not that a rectangular matrix fails the test. It never reaches the test. Everything so far has been about what the definition demands. The honest way to test that is to weaken it and watch what breaks. So take six readings of what it might mean for two matrices to be partners, and run the same four questions at all six. Both products the identity, which is the real one. The left product alone. The right product alone.

6:45The two products merely agreeing with each other. Both products the zero matrix. And one product zero with the other the identity. The four questions are: how many of a hundred and forty-five matrices find a partner, how many of the non-square ones do, how many find more than one, and how many pairs the reading answers differently depending on which way round you ask it. The real definition: forty find a partner, none of them non-square, none with more than one, and no pair read differently either way round.

7:26Now the rival rows, and the first one is the whole argument in a single line. The right product alone: fifty-eight find a partner, eighteen of them non-square, twelve find more than one, and forty-two pairs are read differently each way round. Weaken the definition by half and squareness stops being forced, uniqueness stops holding, and the relation stops running both ways. All three at once. The merely-agreeing reading is more interesting than it looks. Every one of the eighty-one square matrices finds something it agrees with, and every one of them finds more than one.

8:09But it still lets no non-square matrix through, because two products of different orders cannot agree either. So agreeing forces squareness while forcing nothing else. It is the wrong condition for the right reason, which is the hardest kind to spot. The both-zero reading forces nothing at all: all hundred and forty-five find a partner, including all sixty-four non-square ones. The zero matrix is a partner for everything. And the last reading, one product zero and the other the identity, is met by nothing. Zero out of everything.

8:51That one is not a weaker condition. It is an impossible one, and a student who picks it has not chosen a looser definition. They have chosen an empty one. Look again at that last column, the one counting pairs a reading answers differently depending on the order you ask in. For the real definition it is zero. For each of the one-sided readings it is forty-two. That is the both-ways property, measured rather than asserted.

9:26If B is an inverse of A, then A is an inverse of B, and the reason is that the two products in the definition are the same two products read in the other order. Which is why nobody says A has an inverse called B and B has something else. They are inverses of each other. The one-sided readings do not have that. Being a right partner of something is a different property from having one.

9:57Now the result that makes the notation possible. A square matrix has at most one inverse. Suppose it had two. Call the matrix A and the two candidates B and C, each satisfying the definition with A. The proof is a chain of five links, and three separate facts get used along it. Start with B. First link: B equals B times the identity, because that is what the identity does.

10:27Second link: replace that identity with A times C. We are allowed to, because C is an inverse of A, so A times C is the identity. Third link: regroup. B times the quantity A times C becomes the quantity B times A, times C. That is associativity, and it is the only place it is used. Fourth link: B times A is the identity, because B is an inverse of A. So we have the identity times C.

11:00Fifth link: the identity times C is C. And the chain has taken us from B to C. They were the same matrix all along. Notice which halves of which conditions were used. C's condition was used at the second link, with A on the LEFT of C. B's condition was used at the fourth link, with A on the RIGHT of B. Two different halves, from two different candidates. A one-sided definition would have supplied one of them and not the other.

11:34You can see how load-bearing that second link is. Hold A and B fixed, let C range over eighty-one candidate matrices, and ask which links still hold. Four of the five links hold for all eighty-one. The second link holds for exactly one, and that one is the partner we started with. To see that the chain really needs both halves, run it where only one half is available. Here is a matrix with two rows and three columns: one, zero, zero above zero, one, zero.

12:12Search seven hundred and twenty-nine candidates with three rows and two columns for one whose product with it, in that order, is the identity. Nine of them work. Nine different partners for one matrix. They agree completely in their first two rows and differ only in the third, which the product never looks at. So the one-sided reading has just handed us nine inverses of the same matrix, and uniqueness is gone.

12:44How many of those nine satisfy the full definition? None. The other product is three by three and it is never the identity. Now feed one of the nine into the uniqueness chain and watch where it stops. First link holds. Second link holds. Third link, the regrouping, holds. Fourth link fails, and it is exactly the link that reached for the other half of the condition. The proof is not merely unproved without the second demand. It has a specific step that stops working, and you can point at it.

13:24That third link deserves a moment, because it is the one people read past. Regrouping a triple product is a real fact about matrix multiplication, and it is not automatic. Take the sixteen two by twos built from zero and one, and every triple you can make from them. Four thousand and ninety-six triples. Multiply the first two and then the third, then multiply the last two and put the first in front. Compare.

13:56All four thousand and ninety-six agree. Not one disagreement. But that number is only worth something if the same sweep could have found a disagreement. So run it again on an operation that is not associative: take two matrices to the difference of their two products, both ways round. Same population, same triples. Two thousand two hundred and eighty-four of them disagree. So the sweep can fail, and on multiplication it did not. That is what makes the first number a measurement.

14:34One more result, and it will feel familiar if you have met the transpose. If two matrices both have inverses, so does their product, and the inverse of the product turns the two factors around. The inverse of A times B is the inverse of B, times the inverse of A. Reversed, not straight. The proof is a short ladder, and every rung multiplies on the same side. Start from the definition applied to the product: A B times its inverse is the identity.

15:09Multiply on the left by the inverse of A. The A and its inverse collapse and you are left with B times the product's inverse equal to the inverse of A. Multiply on the left by the inverse of B. The B collapses the same way, and what remains is the inverse of the product, equal to the inverse of B times the inverse of A. Every step was a left multiplication. Multiply on the right at any rung and nothing collapses.

15:45Now count it. Of the eighty-one two by twos here, forty have a partner. Pair those forty with each other and you get sixteen hundred ordered pairs. The reversed arrangement is the product's inverse in all sixteen hundred. The straight arrangement is the product's inverse in two hundred and eighty, and wrong in the other thirteen hundred and twenty. Run the same sixteen hundred pairs through the transpose instead. Reversed: sixteen hundred. Straight: two hundred and eighty.

16:21The same two numbers, and pair for pair the two laws agree on exactly which pairs the straight order happens to work for. Zero disagreements. That is not a coincidence. Both laws come from the same place, and once you have one the other costs you almost nothing. A small thing that is worth being pedantic about, because it is the commonest slip in this material. The identity matrix is not the number one.

16:53Write a one by one arrangement holding the single entry one. That IS the identity of its order, and it is a matrix. It is not the two by two identity. And no product of two by twos is ever equal to it: the product of two by twos is always two by two. So a line that sets a product of matrices equal to the digit one is not a shorthand. It is a sentence with nothing on one side of it.

17:25The two things also behave differently. Take a matrix with two rows and three columns. Multiplying it on the LEFT by the two by two identity leaves it unchanged, for all sixty-four in the population. Multiplying it on the RIGHT by the same identity is not defined at all. Zero out of sixty-four. Scaling by the number one, on the other hand, leaves all sixty-four unchanged from either side, because it is not a multiplication of matrices at all.

18:02Same symbol, in speech. Different objects, on the page. So take stock, because what this definition does not give you is as important as what it does. You can now say what an inverse is, verify a candidate in two multiplications, say why a rectangular matrix cannot have one, prove there is at most one, and turn the factors around in a product. What you cannot do is produce one.

18:33Nothing here tells you which matrices have an inverse, and nothing here hands you a method for building it. Count how bad that gap is. Of the eighty-one two by twos built from minus one, zero and one, forty have a partner among the candidates offered. Forty-one do not. The definition is completely silent about which are which. That count came from brute search: try every candidate, do both multiplications, and keep the ones that work.

19:07There is a second, completely different route to the same forty, one that never multiplies a single pair. It agrees, matrix for matrix. That route is the beginning of the method, and it is a topic of its own. For now the honest summary is this. Two products, not one. Both square, both the same order. At most one partner, and the factors turn around. And the question of which matrices have a partner at all is still open.