Deleting a row and a column: minors, and the sign that turns one into a cofactor

0:00Here is the value of a three by three, written out in full. Six terms. Each term is three symbols multiplied together, so eighteen symbol slots in all, and half the terms carry a minus. It is correct, and it is unusable. Not because it is hard, but because there is nothing in it you can point at. Now watch what happens if we give two things a name. One entry, times something. Plus a second entry, times something. Plus a third entry, times something.

0:35Three products added. Six symbol slots instead of eighteen. Nothing has been computed. Not one multiplication has been saved. The same eighteen slots are still in there, hiding inside the somethings. What has changed is that the expression now has parts you can talk about, and once a thing has a name you can state a rule about it. So this is really about naming. Let us make the naming earn its keep.

1:07Pick any entry of a three by three. It sits in exactly one row and exactly one column. Delete both of them. The whole row. The whole column. Nine entries go in; five of them are struck out; four survive. Those four form a two by two, and its value is called the minor of the entry you picked. Say the count out loud, because it is where most of the mistakes live. FIVE go, not one.

1:38The entry you chose is one of the five. It is inside its own row and inside its own column, so it leaves with them. That will matter later, and it is not a detail. Write the minor with a capital M carrying the entry's two subscripts: the row number first, then the column number. And notice what the survivors sit inside. Bars, not brackets. A minor is a determinant, not an array of numbers waiting to be one.

2:11Now, this is a definition, and definitions are the easiest thing in mathematics to nod along to. So we are not going to nod. We are going to run six readings of 'delete something' side by side and see which of them survives. One: delete the entry's own row and its own column. The real one. Two: delete only the entry itself. Three: delete only its row. Four: delete only its column.

2:42Five: delete row j and column i -- the two subscripts read the wrong way round. That is not a silly rule. That is a slip of the pen. Six: delete row i and column i, ignoring the second subscript entirely. And four questions, asked of every one of them, over all five hundred and twelve three by threes built from nought and one. Does what it leaves have a shape -- square, and one order smaller?

3:12Paired with signs and entries, does it give back the right value along the top row? And along all six lines? And this one, which is the sharpest: can any cofactor it produces see the entry it belongs to? The real deletion: five hundred and twelve, five hundred and twelve, five hundred and twelve, five hundred and twelve. All four, everywhere. Deleting only the entry: nought, nought, nought, nought. Deleting only the row: the same four noughts. Only the column: the same.

3:51Those three fail at the first hurdle. Delete one entry from nine and you have eight numbers in a ragged shape, which is not a determinant and cannot be given a value at all. Now the interesting two. Reading the subscripts the wrong way round leaves a perfectly good two by two -- five hundred and twelve for shape. It gets the top row right on three hundred and forty-eight, and all six lines right on only one hundred and seventy-eight.

4:24And the last column, the sharp one, is sixty-four. Sixty-four out of five hundred and twelve. On the other four hundred and forty-eight arrays, at least one of its cofactors CHANGES when you change the very entry it is supposed to belong to. That is the disease, and it is worth being precise about why. If you delete row j and column i, then for any off-diagonal position the entry itself is still sitting there in the survivors. Its cofactor is partly made of it.

5:01The real deletion cannot do that, ever, on any array, because the entry is inside both of the lines that go. So 'delete the row AND the column' is not a convention chosen for tidiness. It is the only reading under which a cofactor is a property of the POSITION rather than of the number sitting in it. The last rival, ignoring the second subscript, is blind in the same way -- five hundred and twelve -- but it gets all six lines right on only two hundred and seven. Blindness alone is not enough.

5:41One row goes and one column goes, so the order drops by exactly one. Always. On all five hundred and twelve three by threes, every one of the nine minors has order two. Run it downward. A four by four gives minors of order three. A three by three gives order two. A two by two gives order one. And a one by one gives nothing at all. Delete its only row and its only column and there is no array left to value.

6:15Which is why the rule is stated for order two and upward, and why the bottom is a genuine edge rather than a fussy exclusion. Now here is something worth an extra thirty seconds, because it explains a formula you learned without explanation. At order two, every minor has order one. A one by one determinant is just the number inside it. So the minor of an entry in a two by two is not something you work out. It IS another entry -- the one diagonally opposite.

6:52Checked on all eighty-one two by twos built from minus one, nought and one: every single minor is exactly the diagonally opposite entry, all eighty-one times. And that is why the two by two rule looks like two products rather than like an expansion. It IS an expansion. Its minors are just so small that nobody notices them. One minor, located and worked. Take the array holding the nine counting numbers in order: one, two, three on top; four, five, six; seven, eight, nine.

7:32Find the entry six. It is in the second row and the third column. So strike out the second row -- four, five, six, all gone -- and the third column -- three, six, nine, all gone. Four survivors: one and two along the top, seven and eight below. Its value is one times eight, minus two times seven. Eight minus fourteen. Minus six. Now, the entry was six and its minor is minus six, and you should hear that and be suspicious rather than pleased.

8:07It is a coincidence. There is no rule here. Nothing about an entry predicts its own minor -- that is the whole content of the last scene. By the way, the array itself has value nought, and nothing in this calculation needed to know that. A minor is a local object. Now the second name, and it is a small one. Take the minor of an entry. Multiply it by plus one or by minus one, according to where the entry sits. What comes out is called the cofactor of that entry, and it gets a capital A with the same two subscripts.

8:49That is the entire definition. A cofactor is a minor with the sign of its position already welded on. The sign is minus one raised to the sum of the two subscripts. So if the two subscripts add to an even number, the cofactor IS the minor. If they add to an odd number, it is the minor's negative. Nothing else ever happens. There is no third case. And the obvious question: why two names for things that differ by a sign?

9:23Because one of them can be added up and one of them cannot. The compact line from the first scene works only if the sign is already inside the object you are multiplying by. The minor is the raw material. The cofactor is the part that fits. Where the sign comes from, position by position. The top row holds positions one-one, one-two and one-three. Their subscript sums are two, three and four.

9:55The second row: three, four, five. The third row: four, five, six. Even, odd, even; odd, even, odd; even, odd, even. Plus, minus, plus; minus, plus, minus; plus, minus, plus. Now count them, because this is where a very common statement goes wrong. FIVE of the nine positions carry a plus. Four carry a minus. Five and four, not four and five. The five are the corners and the middle: one-one, one-three, two-two, three-one and three-three.

10:31At those five the cofactor is the minor. At the other four it is the minor turned over. There is no position where anything else happens. And notice the sign came out of the subscripts alone. Nobody looked at a single number in the array. Same treatment for the sign as we gave the deletion. Six readings, run side by side over the same five hundred and twelve arrays. The parity of the two subscripts added. No sign at all. Plus, minus, plus along every line by step. The parity turned over. The parity of the two subscripts MULTIPLIED. And the parity of the row subscript alone.

11:19Three questions each: right along the top row, right along all six lines, and do the six lines at least agree with each other. The subscript sum: five hundred and twelve, five hundred and twelve, five hundred and twelve. Everything. No sign at all: four hundred and sixteen, three hundred and twenty-four, three hundred and twenty-four. And now the trap. Plus, minus, plus by step: five hundred and twelve on the top row -- perfect -- and three hundred and thirty-eight on all six.

11:56Read that again. There is a rule here that is right on the top row of every single array we tried, and wrong the moment you expand along the second row. It is the rule almost everyone actually carries in their head. Plus, minus, plus. And it is correct exactly where students first meet it, which is why it survives. The parity turned over gets three hundred and thirty-eight both times, but look at its third number: five hundred and twelve. All six of its lines agree with each other, on every array, and all six are wrong together.

12:37Agreement across the six lines is not correctness. It never was. The parity of the product: three hundred and thirty-eight, three hundred and six, four hundred and sixteen. The row subscript alone: three hundred and sixty-four, two hundred and ninety, two hundred and ninety. Two of the six are right along the top row. One of the six is right along all six lines. One more thing about that table, and it is the reason a checker has to ask which and not just how many.

13:14The real rule marks five of the nine positions plus. The parity of the two subscripts MULTIPLIED also marks five of the nine positions plus. Same count. Five and five. So a test that asked 'does your rule put a plus at five positions?' would pass both of them. Now put the two grids side by side. The real rule's five are the four corners and the centre. The product rule's five are the four edge-middles and the centre.

13:46Of the nine positions, the two rules give the same sign at exactly ONE. The centre. Everywhere else they disagree. Two rules, identical by every count you could take, agreeing about one square out of nine. That is why the sign is stated as a parity and not as a pattern to remember, and why counting is never the same thing as identifying. The smallest complete example. A two by two: one and minus two on top, four and three below.

14:23Four entries, so four minors and four cofactors. Eight numbers, and every one of them is a single entry of the array, because the minors have order one. The minors, reading across the top row and then the second: three, four, minus two, one. Now the signs. Top left, subscripts add to two, even, plus. Top right, three, odd, minus. Bottom left, three, odd, minus. Bottom right, four, even, plus.

14:55So the cofactors are three, minus four, two, one. Exactly two of the four changed: the two off the diagonal. The two on the diagonal were left alone. That is the cofactor rule happening in front of you, at the smallest size where it can happen at all. Now the same thing in letters, which is where the two names stop being interchangeable. Take a general three by three and write two cofactors: the one at the top left, and the one directly below it.

15:30Top left. Strike out the first row and the first column; four survivors; the minor is a-two-two times a-three-three, minus a-two-three times a-three-two. Its subscripts add to two. Even. So the cofactor is that expression unchanged. Now the one below. Strike out the SECOND row and the first column. Different four survivors, so a genuinely different minor: a-one-two times a-three-three, minus a-one-three times a-three-two. Its subscripts add to three. Odd. So every term flips: the cofactor is minus a-one-two a-three-three, plus a-one-three a-three-two.

16:18Two things happened there and it is worth separating them. First, the minors are different objects, because different rows were struck out. Second, and on top of that, one of them has been negated. Students who think minor and cofactor are two words for one thing are usually only seeing the second half. And now the payoff, which is the whole reason any of this was worth naming. Go back to the long line: six terms, eighteen symbol slots.

16:52Take the general three by three, expand along the top row, and substitute the cofactor names in wherever they fit. The first entry times its own cofactor. Plus the second entry times its own cofactor. Plus the third entry times its own cofactor. Three products added. Six symbol slots. And it is not an approximation of the long line, or a rearrangement of it. Multiply the short form out in letters and you get the long form back, term for term.

17:28One sentence now says it: the value is what you get by pairing every entry of any one line with its own cofactor, and adding. ANY one line. The freedom to choose which was earned earlier, the long way, and this is where it gets cashed. Take the array with rows one, nought, four; three, five, minus one; nought, one, two. Its value is twenty-three. Along the top row: twenty-three. The second row: twenty-three. The third: twenty-three. All three columns: twenty-three.

18:05Six lines, one value, one sentence. Checked on every one of nineteen thousand six hundred and eighty-three arrays built from minus one, nought and one -- all six lines right, every time. What you will actually be asked to do is grind these out, so let us grind some out. A two by two with two and minus four on top, nought and three below. Minors: three, nought, minus four, two. Cofactors: three, nought, four, two.

18:38Look at the second one. Its position is top right, subscripts add to three, odd, so the rule attaches a minus -- and the minor is nought, so nothing visible happens. The sign was still attached. It was decided before anybody looked at the number, and nought is the one value that hides it. The identity array makes that point four times over. Its nine minors are one, nought, nought; nought, one, nought; nought, nought, one -- and its nine cofactors are exactly the same nine numbers.

19:16All four of its sign flips land on a nought. Four invisible minus signs, every one of them genuinely there. Now a full one. Rows one, nought, four; three, five, minus one; nought, one, two. Nine minors: eleven, six, three; minus four, two, one; minus twenty, minus thirteen, five. Nine cofactors: eleven, minus six, three; four, two, minus one; minus twenty, thirteen, five. Four of the nine changed, and they are the four positions whose subscripts add to an odd number. The same four, every time, on every array there has ever been.

20:00Two three by threes like that is eighteen minors and eighteen cofactors, and the grinding is the point: the rule is short enough that the only way to get it wrong is to lose track of where you are. So what did naming buy? The minor: strike out the entry's own row and its own column, and value what is left. Five of nine gone, four left, order down by one.

20:29The cofactor: that minor with the sign of its position welded on. Even subscript sum, unchanged. Odd, turned over. Five positions and four. And with those two names, the value of a three by three is three products added, along any line you like. Nothing was computed that could not have been computed before. The eighteen slots are all still there. What changed is that a thing with a name can appear in a sentence, and that sentence is short enough to be true of every order at once.

21:07Which is the actual reason this is worth ninety seconds of anybody's time: the compact form is what the next idea is going to be built out of. You cannot state a rule about all nine cofactors of an array until the nine cofactors are objects with names. That is what the naming did. It turned nine numbers into nine named things.