The determinant: signed area

How much does a map stretch area?

Take a shape, apply a linear map, and compare its area before and after. For a general map the answer depends on the shape. For a linear map it does not: every region is scaled by the same factor, so it is enough to watch one region — the unit square. Its image is a parallelogram whose area is the scale factor, and the signed area of that parallelogram, with a sign tracking orientation, is the determinant.

That definition buys a lot. The determinant tells you whether a map stretches or shrinks, whether it flips the plane, and — most importantly — whether it can be reversed. A map that multiplies area by zero has crushed the plane onto something thinner than a plane, and once information is destroyed it cannot be recovered.

Because the determinant is an area ratio, it behaves the way ratios should: compose two maps and their area factors multiply. If A scales every area by det A and B scales every area by det B, then doing B and then A scales every area by the product, so

This is the bridge back to Part 4. Composition of transformations multiplies matrices, and the one number attached to each matrix — its determinant — multiplies along with them. It is the cleanest example in the series of a geometric operation on maps turning into a simple arithmetic rule on numbers.

A word on the word "signed". Ordinary area is a positive number, but the determinant carries a sign because a linear map can do something an area calculation alone cannot see: it can turn the plane over. Picture the plane as a sheet with the standard axes drawn on it, the horizontal axis first and the vertical second. A map that preserves the counterclockwise order of those axes has a positive determinant; a map that reverses it has a negative one. The magnitude is how much areas grow, and the sign is which side of the sheet you are now looking at. Both pieces of information live in the same number.

The unit square becomes a parallelogram

The unit square has corners (0,0), (1,0), (1,1), (0,1) and area 1. A linear map sends the two basis vectors to the columns of its matrix, so it sends the square to the parallelogram spanned by those two columns. Its area is the magnitude of the determinant, and its signed area is the determinant itself:

Because every small piece of the plane is scaled by the same factor, that ratio is not special to the square. A circle, a blob, a picture of a cat — after the map each has its area multiplied by |det A|. Edit the matrix below and watch the parallelogram, its shaded area, and the readout move together. When the determinant is negative the fill turns to the warning colour and the plane has been flipped over, like a page turned face down.

Two facts fall straight out of the picture. Swapping the two columns swaps the two spanning vectors and reverses the winding of the parallelogram, which flips the sign of the determinant — orientation is why the determinant is antisymmetric. Duplicating a column leaves the two vectors parallel, so the parallelogram has no area and the determinant is zero.

Written out in coordinates, the signed area of the parallelogram spanned by the columns of A is the familiar two-by-two formula:

The first term ad is the area of one rectangle and bc is the other; the subtraction is the correction that turns the pair into the slanted parallelogram, and its sign tells you which way the second column lies relative to the first. If that is the only determinant formula you remember, the picture still lets you reason about every case: ad - bc = 0 means the two columns are parallel, and the parallelogram has flattened to a segment.

You can also read the size of the determinant as a measure of how far the matrix is from a collapse. A determinant of 2 means the map doubles every area; a determinant of 0.5 means it halves them. A determinant of -2 still doubles areas, but it also flips the plane, so the image parallelogram is traversed the other way round. Try the flip preset: the shape and size of the parallelogram are unchanged from the identity's, only the fill colour and the sign reveal that orientation reversed.

The same picture explains how the determinant behaves under the operations you already know. Multiplying the whole matrix by a scalar scales both columns and therefore the area by that scalar squared in two dimensions, which is why \det(kA) = k^2 \det A here. Adding a multiple of one column to the other slides one spanning edge parallel to itself and leaves the area unchanged. Those facts, together with the sign flip when two columns swap, are enough to derive every rule the determinant obeys.

A one-slider family, from identity to ruin

Watch a single parameter drive a matrix from the identity to a full collapse. Start at the identity, where the determinant is one and the parallelogram is the unit square. As the slider grows, the second column slides toward the first; the parallelogram tilts, then flattens, then at the end the two columns are the same vector and the image is a bare line segment with no area at all.

At that endpoint the determinant is zero and the map has destroyed a whole dimension of information. Two different points on the plane land on the same point of the line, so there is no way to run the map backwards: a single output no longer identifies its input. The readout calls it by name — information destroyed.

The set of points that get squashed to zero has a name of its own: the null space of the matrix. For the collapsing family here it is a whole line of inputs that all map to the origin, and its dimension is exactly the number of directions the map threw away. A matrix with a nonzero determinant has a null space containing only the origin, because nothing except zero is sent to zero. A matrix with determinant zero always has a null space of positive dimension, and that is the precise sense in which it is not invertible: several inputs share each output, so no rule can tell them apart.

Computing a determinant by hand is easiest for triangular matrices. If every entry below the diagonal is zero, the unit cube becomes a slanted box whose volume is just the product of the diagonal entries, so the determinant is that product. Elimination reaches a triangular form by row operations that leave the determinant unchanged apart from tracking row swaps, which is exactly why practical determinant routines reduce first and multiply the pivots afterwards. The permutation-and-cofactor formulas you may have memorised are the same volume count written in a different order.

Undoing the map, or proving there is no undo

Undoing a linear map means finding a second map that puts everything back. That second map is the inverse matrix, and it exists precisely when the first map is reversible — which, in area terms, is exactly when it did not flatten the plane. So the test is one number: the determinant. If it is nonzero there is an inverse; if it is zero there is none.

The demo computes the inverse with Gaussian elimination and shows it whenever the determinant allows. Undoing the map and then doing it, or doing it and then undoing it, returns the identity grid — the product A^-1A is the identity matrix to numerical precision. Push the determinant to zero and the inverse simply ceases to exist; the readout says so plainly.

This is the first appearance of a theme the series returns to again and again: a geometric question — can I undo this? — collapses to a single algebraic number you can compute. Rank, null space, and the four subspaces are the full version of the same story, and they all start from the fact that a zero determinant means a dimension was lost.

The inverse carries the area factor with it, inverted. If A scales area by det A, then undoing A must scale area by the reciprocal, so det(A^{-1}) = 1/\det A. That is why a determinant close to zero is dangerous even when it is not exactly zero: the inverse's area factor blows up, and with it any error that entered through the input. The demo shows the matrix of the inverse, but the number that governs how much to trust it is the determinant of the original.

For a two-by-two matrix the inverse has an explicit formula that makes the connection to area unmistakable. Swapping the diagonal entries and negating the off-diagonal ones produces what is called the adjugate, and dividing by the determinant produces the inverse:

The division is the whole story. When ad-bc is nonzero the inverse is a perfectly ordinary matrix; as the determinant shrinks toward zero the entries of the inverse grow without bound, and at zero the division is undefined and no inverse exists. The determinant does not merely detect invertibility; it measures how comfortably invertible a matrix is.

This also tells you how to solve a square system Ax = b: apply the inverse, so x = A^{-1}b. The formula is correct and worth understanding, but it is almost never the way a computer should solve the system. Forming the inverse costs more work and introduces more rounding error than eliminating directly, and if the determinant is small the computed inverse is garbage. The determinant's job here is diagnosis — is the system even solvable, and how trustworthy is the answer — while elimination, in the next part, is the cure.

For a diagonal matrix the determinant is just the product of the diagonal entries, because the map stretches each axis independently and the unit square becomes an axis-aligned rectangle. That special case is the picture behind elimination: row operations reduce a matrix to triangular form, the determinant becomes the product of the pivots, and the only care needed is tracking how many row swaps happened along the way.

The same number, one dimension up

The idea does not stop in the plane. For a 3×3 matrix the determinant is the signed volume of the box that the unit cube becomes, and for an n×n matrix it is the signed n-dimensional volume of the image of the unit hypercube. The bookkeeping is identical: absolute value is the size of the stretch, sign is orientation, and zero means the cube was squashed flat onto a lower-dimensional slab.

Edit the 3×3 matrix below. The faint box is the unit cube, drawn in an oblique projection so a flat page can show depth; the highlighted box is its image under the map. The determinant is computed directly from the matrix, and its absolute value is the factor by which every volume is scaled.

This is the definition that survives all the way to determinant formulas built from permutations and cofactors. Those formulas look like combinatorial trivia until you remember what the number means: a volume whose sign says which way the space is wound.

The area rule generalises without change. The determinant is multiplicative in every dimension, det(AB) = det(A)det(B), so a chain of maps scales n-dimensional volume by the product of their determinants. It is also why a determinant of zero means a collapse regardless of dimension: the image of the cube loses all its volume, which can only happen if the map sent n-dimensional space into something of lower dimension. In the plane that lower-dimensional image is a line or a point; in three dimensions it is a plane, a line, or a point. In every case, a direction has been folded away.

In three dimensions the determinant also has a tidy expression in terms of the columns. Treat the three columns as vectors and take the dot product of the first with the cross product of the other two; that number is the signed volume of the parallelepiped they span, and it equals the determinant. The cross product, the volume, and the orientation all come from the same construction, which is why the determinant shows up wherever a normal vector or a handedness is needed — the cross product and its skew-symmetric matrix are the subject of a later part.

It helps to see the determinant as one idea rediscovered rather than a new concept per dimension. In one dimension a linear map is multiplication by a number, and the determinant is that number: the factor by which length scales. In the plane it is the factor by which area scales; in space, volume. Each time the space grows, the determinant is the single number that measures the stretch in all directions at once, and the pattern 0 meaning collapse never changes.

One number, two worlds

The determinant keeps reappearing because it is the number that says whether a linear map is safe to undo. It is zero exactly when the columns are dependent, exactly when the rank drops below the number of columns, exactly when the null space is bigger than the origin, and exactly when zero is an eigenvalue. All of those are the same statement wearing different notation, and later parts choose whichever version makes the picture clearest. It also equals the product of the eigenvalues, which is why a matrix with a zero eigenvalue is the collapsing case you have just watched.

Wherever a coordinate change has to be accounted for, a determinant does the accounting: the absolute value is the local volume factor, and the sign is whether the change preserved local orientation. That is the content of the change-of-variables formula in multivariable calculus, and it is the reason optimization and physics track determinants when they transform coordinates. Keeping the geometric reading in mind turns those formulas into bookkeeping you can follow rather than facts to accept.

Further reading

The references below treat the determinant as a volume first and a formula second, which is the order that makes it memorable. If you take away one thing, take away the picture of the unit square becoming a signed parallelogram.

Strang's Lecture 18 is the best place to see the ten properties of the determinant derived from that picture, including the product rule and the behaviour under row swaps. The Immersive Math chapter makes the signed area draggable, and Axler reverses the usual order by defining the determinant as a volume and deriving the formula from it, which is worth reading once you are comfortable with the geometric version.

• Grant Sanderson, "The determinant", Essence of Linear Algebra, 3Blue1Brown — the chapter this part follows.
• Gilbert Strang, 18.06 Linear Algebra, MIT OpenCourseWare — Lecture 18, properties of determinants.
• Immersive Math, Chapter 5: Determinants — signed area and volume you can drag.
• Sheldon Axler, Linear Algebra Done Right, chapter 10 — the determinant derived from volume rather than the other way round.

Cheat sheet