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

0:00You have met the inverse of an array before. A second array that multiplies with the first, in either order, and gives the identity. What you were never told is which arrays have one. That is a strange gap to leave open. You can define a thing perfectly well and still have no way of deciding whether any particular case exists. Today it closes, and it closes with one number.

0:29Not a procedure. Not a search. One number, computed once, and the answer is settled either way. Two words first, because the answer needs somewhere to put things. An array whose determinant is nought is called singular. An array whose determinant is not nought is called non-singular. That is the whole of both definitions. One number decides which word you use. Here are two arrays that make the point. One and two across the top, four and eight below. And one and two across the top, three and four below.

1:10They differ in two entries. Nothing else. The first has determinant eight minus eight, which is nought. Singular. The second has determinant four minus six, which is minus two. Non-singular. So the line between the two cases can run between two arrays that look almost the same. You cannot eyeball it. That last sentence deserves testing rather than asserting, so let us test it. Take every three by three you can build out of noughts and ones. Five hundred and twelve arrays.

1:47For each one, decide whether it really has an inverse - by row reduction, which computes no determinant and forms no cofactor at all. It either finds an inverse or the rows collapse. A hundred and seventy-four of the five hundred and twelve have one. Now line up seven ways somebody might try to guess that answer, and score each of them against the truth. The determinant is not nought. No entry is nought. The determinant is not one. The array is not symmetric. The diagonal does not add to nought. No row is all noughts. The determinant is positive.

2:31Each one is asked three things. How many arrays does it accept? How many does it accept that have no inverse? And how many does it turn away that do? Here is the table. The determinant test: accepts a hundred and seventy-four, accepts nothing it shouldn't, turns away nothing it shouldn't. Perfect. No entry is nought: accepts exactly one array out of five hundred and twelve - the one that is all ones - and that array is singular. It turns away all a hundred and seventy-four. It is not merely wrong. It is almost exactly backwards.

3:14The determinant is not one: accepts four hundred and twenty-eight, three hundred and thirty-eight of which have no inverse. The array is not symmetric: four hundred and forty-eight accepted, three hundred and six of them wrongly. No row is all noughts: three hundred and forty-three accepted, a hundred and sixty-nine of them wrongly - though notice it never turns away a good one. It is a necessary condition that is nowhere near sufficient.

3:48One test out of the seven is exactly right, at order three and again over all eighty-one two by twos. And it is the one the definitions are built on. One more row is worth a moment, because it is the subtle failure. The determinant is POSITIVE. That test never once accepts an array with no inverse. Not on the five hundred and twelve, not on the eighty-one. If you only counted false accepts, it would look perfect.

4:22But it turns away eighty-seven invertible arrays at order three and twenty-four at order two - every array whose determinant is negative. So a test can be flawless in one direction and useless in the other. Getting no false positives is half a question. And two of the seven are stranger still. Not symmetric and diagonal does not add to nought accept exactly the same NUMBER of arrays at both orders - four hundred and forty-eight, and fifty-four.

4:57They are not the same arrays. Fifty-six arrays meet one and not the other, each way round, at order three. Same score, different sets. A tally is not an identification. So here is the statement everything so far has been building towards. A square array has an inverse exactly when its determinant is not nought. Read the word EXACTLY carefully. It is two claims in one. If an inverse exists then the determinant is not nought. And if the determinant is not nought then an inverse exists.

5:38That second half is the one that does real work, because it hands you the inverse rather than merely promising it. And the first half is what tells you when to stop looking. If the determinant is nought, there is nothing to find. Not hard to find. Not there. Both halves get proved. Let us do the easy one first. Suppose the array has an inverse. Call it B. Then A times B is the identity. Take determinants of both sides.

6:12On the right, the determinant of the identity is one. On the left - and here we borrow something - the determinant of a product is the product of the two determinants. So the left is the determinant of A times the determinant of B. Two numbers multiplying to one. Neither of them can be nought, because nothing times nought is one. So the determinant of A is not nought. Three lines, and we are done.

6:44I want to stop on that borrowed step, because it is the only thing holding this half up and it was not proved anywhere. The determinant of a product is the product of the determinants. It sounds obvious. It is not obvious, and multiplication of arrays is exactly the sort of thing that does not respect nice rules. So let us at least measure it. Take all eighty-one two by twos built from minus one, nought and one, and every ordered pair of them. Six thousand five hundred and sixty-one pairs.

7:21The product rule is right on all six thousand five hundred and sixty-one. And three rivals are not. Add the two determinants instead: right on one thousand one hundred and five. Take the first one alone: three thousand six hundred and thirty-three. Take whichever is bigger: two thousand two hundred and nine. That last one is right more than a third of the time, which is exactly how a wrong rule survives a spot check.

7:53Measured, not proved. I am telling you which it is. Now the half that matters. Suppose the determinant is not nought. Build an inverse. Start from the thing we already have. The array times its adjoint is the determinant times the identity. The determinant is a number, and it is not nought. So we may divide through by it. The array, times the adjoint over the determinant, equals the identity.

8:25And the same the other way round, because the adjoint result held in both orders. Now read that line again and ask what it says. Something multiplies A, on either side, and gives the identity. That is not evidence for an inverse. That is the definition of one. The inverse is the adjoint over the determinant. Go back over that argument and find the place where the condition was used.

8:56There is exactly one. The division. Every cofactor you have computed, every minor, the whole flip that builds the adjoint - all of it was to get an equation you could divide. If the determinant is nought, nothing in the construction breaks. The adjoint still exists. The equation still holds - with nought on the right. The only thing that fails is the one step you needed. That is why singular arrays have adjoints and no inverses. Nothing is missing. One division is illegal.

9:34The formula is short enough to misremember, so let us do to it what we did to the test. Six ways to build a second array out of the pieces on the board. The adjoint over the determinant. The adjoint on its own. The cofactors over the determinant, without the flip. The array itself over the determinant. The adjoint multiplied BY the determinant. The transpose over the determinant. Run each one over the hundred and seventy-four invertible three by threes and ask a single question: is it the array row reduction found?

10:15The adjoint over the determinant: a hundred and seventy-four out of a hundred and seventy-four. The cofactors without the flip: thirty-two. The array over the determinant: one. The transpose over the determinant: three. Those three are simply wrong and you would notice. The other two are the interesting ones. The adjoint on its own, with no division at all, is the inverse on eighty-four of the hundred and seventy-four. Which eighty-four? Exactly the arrays whose determinant is one. Of course - dividing by one changes nothing.

10:57And the adjoint MULTIPLIED by the determinant is the inverse on a hundred and sixty-eight of the hundred and seventy-four. That is the arrays whose determinant is one or minus one, where multiplying and dividing do the same thing. A hundred and sixty-eight out of a hundred and seventy-four. That rival is right ninety-six times in a hundred on this collection. So widen the collection. Over nineteen thousand six hundred and eighty-three arrays built from minus one, nought and one, eleven thousand eight hundred and eight have inverses. The real formula gets all eleven thousand eight hundred and eight. Multiplying instead of dividing gets six thousand nine hundred and sixty.

11:50It goes from almost always right to barely more than half. The rival did not change. The test did. Let us invert one properly. Here is a three by three. Nine cofactors, transposed, gives the adjoint - seven, minus three, minus three; minus one, one, nought; minus one, nought, one. Its determinant is one. So the inverse is that adjoint divided by one, which is that adjoint. And now be suspicious of me. The example that teaches the formula is the one example in which the formula's only difficult step is invisible.

12:34You cannot see a division by one. If you learn the shape from this array you will learn the adjoint and forget the divisor. Take the other one instead. One and two on top, three and four below. Determinant minus two. Adjoint four, minus two; minus three, one. Divide, and the inverse has a fraction in it - minus two and one on top, three halves and minus a half below.

13:02That is what an inverse usually looks like. Fractions. The clean one was the exception. Two arrays now, and a rule that everybody memorises and almost nobody is shown. The inverse of a product is the product of the inverses - in the reverse order. Why reversed? Cancel from the inside out. Take A times B, and multiply it on the right by B inverse times A inverse. The inner pair meet first. B times B inverse is the identity, and it vanishes. Now A meets A inverse, and that vanishes too. The identity is all that is left.

13:46Try it the other way and the inner pair is B against A inverse. Those two have nothing to do with each other. Nothing cancels. And here is why this is worth a rule rather than a habit. Over all two thousand three hundred and four ordered pairs of invertible two by twos in our sweep, the reversed order is right on all two thousand three hundred and four. The forward order is right on three hundred and fifty-two of them - exactly the pairs whose inverses happen to commute. It is wrong on one thousand nine hundred and fifty-two.

14:25So it works often enough that one lucky example would have convinced you, and fails on five pairs in six. There is one more route to an inverse and it computes no cofactor whatsoever. Suppose somebody hands you an equation the array satisfies. Here is a two by two: two and three on top, one and two below. Square it and you get seven and twelve on top, four and seven below. Now subtract four times the array, and add the identity.

15:00Everything cancels. You get the array of noughts. So the array squared, minus four times the array, plus the identity, is nothing. Rearrange. Move the identity across: the array times four times the identity minus the array, equals the identity. Read that. Something times A gives the identity - so that something is the inverse. Four times the identity, minus the array. Two and minus three on top, minus one and two below.

15:35Not one cofactor was computed. And it agrees exactly with what row reduction finds. There is a version of that trick that runs backwards, and it catches people out. Instead of handing you the equation, it hands you the array and the SHAPE of the equation, and asks for the missing numbers. Three and two on top, one and one below. Find the two numbers that make the array squared, plus the first number times the array, plus the second number times the identity, come to nothing.

16:12Square it: eleven and eight on top, four and three below. Now match entries. The top right gives eight plus two of the first number equals nought, so the first number is minus four. Then the top left gives eleven minus twelve plus the second number equals nought, so the second number is one. Minus four and one. And once you have them you are back in the forward problem and the inverse falls out the same way.

16:45Three consequences, quickly, because each one is worth thirty seconds. First. What is the determinant of an inverse? Take determinants of the defining equation and you get the reciprocal. We checked that on every invertible array of all three sweeps - a hundred and seventy-four, forty-eight and eleven thousand eight hundred and eight - and it held every time. Saying it is the determinant itself is right on six thousand nine hundred and sixty of the eleven thousand eight hundred and eight, which is not the same thing.

17:21Second. Invert an inverse and you get back where you started. On all a hundred and seventy-four, and on all eleven thousand eight hundred and eight. Third, and this one is a small delight. Some arrays are their OWN inverse. Take one with a single one in the corner and a two by two block of cosine and sine underneath - cosine, sine on one row, sine, minus cosine on the next.

17:50Multiply it by itself and every entry off the diagonal cancels, and the diagonal fills with cosine squared plus sine squared. Which is one. So the array squared is the identity, and the array is its own inverse. No angle was ever chosen. That works for every angle at once. So where does that leave you? Given any square array, one computation tells you whether it can be undone. Determinant nought: singular, no inverse, stop. Determinant not nought: non-singular, and here is the inverse - the adjoint over that number.

18:34You know which single step needed the condition, and you know that everything else in the construction survives without it. You know the reversal rule and why it reverses, rather than only that it does. And you know one route that skips cofactors entirely when somebody hands you an equation. One last thing, said plainly. The half of the proof that hands you the inverse rests on the adjoint result, which was checked at order three rather than proved in general, and the other half rests on the product rule, which was not proved at all.

19:12That is an honest place to stand. You know exactly what is load-bearing, and you know that every count on this board was measured rather than asserted.