The number that says how much room is left

A two-by-two matrix has four entries, and there is a number built from them that turns up in every part of the subject: in whether a system of equations has a solution, in whether a map can be undone, in changing variables inside an integral, in the characteristic polynomial whose roots are the eigenvalues. It is usually introduced as a formula — the product of one diagonal minus the product of the other — and then justified by what it does.

It is a great deal simpler than that. It is an area.

The two numbers in that figure come from computations with no line in common. One walks round the drawn quadrilateral adding cross products of consecutive corners; the other multiplies two pairs of matrix entries and subtracts. They agree, and the assertion that they agree is what the build would stop on.

Why the area is that particular combination #

The parallelogram’s corners are easy to name. The origin stays at the origin. The corner at one east goes to the first column of the matrix; the corner at one north goes to the second; the far corner, being the sum of the other two, goes to the sum of the columns, because that is what a linear map does.

So the parallelogram is spanned by the two columns, and its area is the base times the height. Take the first column as the base. The height is how far the second column reaches perpendicular to it — which is the second column’s component in the direction perpendicular to the first, and that perpendicular component is exactly what the cross-product-like expression ad − bc measures. It is the dot product with one vector turned a quarter turn first.

There is a second derivation, worth having because it generalises and the first does not. Slide the top edge of the parallelogram along its own direction: the base is unchanged, the height is unchanged, so the area is unchanged. In matrix terms, adding a multiple of one column to the other leaves the area alone. Do that repeatedly and any matrix can be brought to a diagonal one, where the area is obviously the product of the two entries. That is Gaussian elimination, and it says that the determinant of a big matrix can be computed the same way — by making the shape simpler without changing the area.

There is a third way to arrive at the same expression, and it is the one that explains why the formula looks the way it does rather than merely confirming that it works. Insist on three things about a function of two vectors: that it is zero when they are equal, that it adds when either argument is split into a sum, and that it gives 1 on the two basis vectors. Those three demands have exactly one solution, and the solution is ad − bc. The first demand is “a squashed parallelogram has no area”, the second is “areas add when a shape is cut”, the third is “the unit square is the unit”. Everything else about determinants is downstream of those three sentences, in every dimension, and the two-by-two formula is what they force when there are two arguments.

The sign, and what it is recording #

The shoelace formula returns a signed number, and the sign is not noise. It records whether the corners, taken in the order the map produces, run anticlockwise or clockwise. A map with a negative determinant has turned the plane over.

This is why a reflection has determinant −1 and a rotation has +1, and why the two cannot be confused by any amount of stretching. It is also why the eight motions of a square split into two halves of four: the four turns and four flips are exactly the motions of determinant +1 and −1. The sign is a two-valued invariant, it is the simplest one in the subject, and it is the same object as the orientation that a Möbius band destroys.

Size and sign are genuinely separate facts, and the figure keeps them separate: the two matrices above have parallelograms of the same area, and no measurement of area alone tells them apart.

Zero, and what a collapse looks like #

A determinant of zero means the parallelogram has no area, which means the two columns point along the same line, which means the whole plane has been squashed onto that line. Everything that follows is immediate rather than a separate theorem:

The map cannot be undone, because two different starting points land on top of one another and nothing can tell them apart afterwards. The system of equations it represents either has no solution or has infinitely many, never exactly one, because the target either is on the line or is not. And whether a matrix is invertible is decidable by computing one number and comparing it against zero.

The shear panel deserves a second look. Its determinant is 1: the square becomes a leaning parallelogram of the same area, which is the ordinary schoolroom fact that a parallelogram’s area does not depend on its lean. Every shear has determinant 1, and shears are the moves that Gaussian elimination is made of, which is the reason elimination does not change the answer.

Areas multiply #

The determinant of a product is the product of the determinants. Stated as algebra it is a small identity that has to be checked entry by entry, and checking it is a page of arithmetic with eight terms that cancel in pairs.

Stated as areas it is a sentence. The first map multiplies every area by its determinant. The second multiplies every area by its own. Doing them in turn multiplies every area by the product. There is nothing left to prove.

That is the pattern this whole essay is an instance of: the algebra of determinants is a list of facts, each of which requires work, and every one of them is obvious about areas. The determinant of a matrix and its transpose agree, because the parallelogram spanned by the rows and the one spanned by the columns have the same area. Swapping two columns flips the sign, because it reverses the walk round the corners. Multiplying a column by a number multiplies the determinant by it, because it stretches the shape one way.

Why one square settles every shape #

There is a gap in everything above, and it is the gap that makes the determinant a fact about the map rather than a fact about one square. The figure measures what happens to the unit square. The claim in use everywhere else is that the same factor applies to every region — that a circle, a triangle or a coastline all have their areas multiplied by the same number.

That does not follow from one measurement, and the argument for it is the one this site keeps returning to. Any region can be approximated from inside and outside by a grid of small squares, as closely as wanted; that is exactly the construction in adding up rectangles. A linear map sends each small square to a small parallelogram of the same shape as the image of the unit square, scaled down — because scaling and translating commute with a linear map — so each little area is multiplied by the same factor. Add them up, refine the grid, and the whole region’s area is multiplied by that factor too.

Two conditions are doing work in that paragraph and both are worth naming. The map must be linear, so that the image of a small square anywhere is a translated copy of the image of a small square at the origin; a map that bends would fail this and the factor would vary from place to place. And the region must have an area to begin with — which sounds like nothing until the function no rectangles settle on is remembered, and then it sounds like a condition.

How rare a collapse is #

A determinant of zero is a condition on four numbers, so the matrices that have it form a thin set: a surface sitting inside the four-dimensional space of all two-by-two matrices, of no volume at all. Pick a matrix by throwing four darts and it will be invertible, with probability one.

Which makes the collapse sound like a curiosity, and it is not, for a reason that is invisible from the yes-or-no question. A determinant close to zero is not a bit like a determinant that is zero; it is nearly as bad. A map whose determinant is a thousandth squashes areas by a thousand, and undoing it multiplies everything — including any error in the numbers being undone — by a thousand. The matrix is invertible and the inversion is useless.

So the honest quantity is not whether the determinant is zero but how close the map is to one that collapses, and that turns out not to be measurable from the determinant at all. A map that halves both directions has determinant a quarter and is perfectly well behaved; a map that doubles one direction and eighths the other has the same determinant and is four times worse conditioned. The number that answers the real question is the ratio of the largest stretch to the smallest, which is a different quantity with a different picture, and it belongs to arithmetic at scale rather than to area.

The determinant and the directions the map leaves alone #

If a map has two directions it does not turn — invariant directions — then in the coordinate system built from them the map is a pure stretch, by one factor along one direction and another factor along the other. A rectangle with sides along those directions is stretched by both factors, so its area is multiplied by their product.

The determinant is therefore the product of the eigenvalues, and this is the same argument as before with a different rectangle chosen. In the figure the two stretches are 3 and 1, and the determinant is 3.

The converse fails, usefully. A determinant of 1 does not mean the map is the identity or even close to it: the shear has stretch factors 1 and 1 and moves nearly every point a long way. Nor does a determinant of 1 mean lengths are preserved — a map that doubles one direction and halves the other leaves every area alone while sending a circle to a very long ellipse. Area is one number, and one number cannot describe a map.

What it costs to work out #

In two dimensions the determinant costs two multiplications and a subtraction. In three it costs a dozen or so. In n dimensions the definition as a sum over all orderings of the columns has as many terms as there are permutations — twenty-four terms at four, three and a half million at ten, and more terms at twenty than there are seconds in the age of the universe.

Nobody computes it that way. The shear argument above is the escape: elimination brings the matrix to triangular form using moves that do not change the determinant, and a triangular matrix’s determinant is the product of its diagonal — which is itself an area fact, since a triangular matrix sends the unit square to a parallelogram with one side along an axis, and its area is base times height with nothing to compute. That costs on the order of n³ arithmetic operations rather than n!, which at twenty is eight thousand against two and a half billion billion.

The distinction between a definition and a method is one of the few places where this site’s subject and its neighbours meet directly: what elimination costs, how the error behaves and when the pivoting matters are questions for numerical linear algebra, and they are questions about arithmetic at scale rather than about area. What belongs here is the reason the shortcut is allowed, and that reason is a fact about parallelograms.

What the picture cannot show #

Two things, and the second is the more serious.

The formula’s own shape is invisible here. In three dimensions the determinant is a sum of six terms, three added and three subtracted, and the pattern of signs is not arbitrary: a term is added when the ordering of columns it uses can be reached from the standard one by an even number of swaps. That is a statement about permutations and their parity, and it is the reason a determinant in n dimensions has n! terms with half of them negative. None of that is legible in a picture of a parallelogram; the picture shows what the number is, and the sum over orderings shows what it costs to write down.

The picture is two-dimensional and the subject is not. In three dimensions the determinant is the volume of the box spanned by three columns, and the sign records whether the three form a right-handed or a left-handed triple. Everything above survives the translation. Beyond three there is no picture at all, and the phrase “the volume it multiplies by” becomes a definition rather than a description — which is exactly the point at which the algebraic properties have to carry the weight the drawings were carrying here.

And the picture shows a determinant, not a matrix. Two very different maps can have the same determinant, as the shear and the identity do. A number that compresses a four-entry object into one entry is throwing away three numbers’ worth of information, and the interesting question is always which three. The determinant keeps whether the map is invertible and whether it turns the plane over, and it discards everything about direction — which is why the invariant directions needed their own essay and their own picture.

Where the ladder goes next #

The determinant answers one question about a map completely and every other question not at all. The rungs above it are the ones that ask what else a single number can be made to carry.

The trace — the sum of the diagonal — is the other number that survives a change of coordinates, and between them the two determine the characteristic polynomial and so the eigenvalues, in two dimensions. That is as far as two numbers go: at three by three there is a third such quantity, at n by n there are n, and the sequence of them is the coefficients of one polynomial.

There is also a direction that leaves matrices behind. The determinant appeared here as the area of an image, which makes it the local stretching factor of a map — and for a map that is not linear, the same number appears as the thing that has to be inserted when changing variables inside an integral, because a small square there is a small parallelogram here. The picture that argument needs is a small square, magnified, and that is a limit rather than a drawing. It belongs with the derivative, which is the same construction one dimension down.