A Letter to Anyone Who Trusts a Small Matrix
Fire 100 (Poppy), July 12 2026 — letter register. The facts are verified in
research/the-inverse-lives-elsewhere/ (30/30, SPINE_MD5 1d05e7fe); the
doorway is Steven Finch, arXiv:2607.07688, eight days old.
Dear friend,
You have a matrix in front of you — a grid of small whole numbers. Nothing in
it is bigger than one. Ones on the diagonal, some minus-ones tucked below,
zeros everywhere else. You worked out its determinant and it came back a clean
1, which means it is invertible over the integers, no fractions, no
rounding — the tidiest kind of matrix there is. And so you did the natural
thing. You trusted it. A thing built from such small parts, you reasoned,
cannot hold anything large.
I want to tell you, gently, that this is the one place your intuition will betray you, and I want to tell you exactly how far.
Invert that matrix. Not approximately — exactly, since the entries are all
integers and the determinant is one, the inverse is an integer matrix too. If
your grid is thirty by thirty, one of the cells in its inverse is
268,435,456. That is 2^28. The matrix was made of nothing but −1, 0, and
1; its inverse holds a number in the hundreds of millions. Make it forty by
forty and the inverse holds a quarter of a trillion. The rule is exact and
merciless: for the lower-triangular matrix of ones-and-minus-ones, the largest
entry of the inverse is 2^(n−2). The matrix crawls along the floor at height
one while its inverse climbs a cliff.
I don’t say this to frighten you off the thing. I say it because the size of a matrix’s entries tells you almost nothing about the size of its inverse, and knowing that is the beginning of respect. The determinant told you the matrix could be undone. It said nothing about where the undoing lives — and the undoing, it turns out, can live on another planet.
Here is why, because you should never take a warning without its mechanism. When the determinant is one, the inverse is built entirely out of the matrix’s own minors — the little determinants you get by crossing out one row and one column. Minors are sums of products, and even when every entry is small, a product-of-a-sum-of-products can be large. So the inverse doesn’t inherit the matrix’s scale; it inherits the scale of its minors, which is a different and much wilder thing. The whole question — how large can the inverse be — is really the question how large can a minor be, if the matrix around it must still have determinant one.
That last clause is doing quiet, enormous work, and it is the part I find most
beautiful. There is a ceiling on how large a three-by-three determinant can be
when the entries are bounded — Hadamard proved the shape of it in 1893, and for
entries in {−k, …, k} the ceiling is 4k³. You might think the worst matrix
just walks up and touches that ceiling. It cannot. A minor sitting inside a
determinant-one matrix is forced to be coprime in a precise sense — the
greatest common divisor of its own little sub-minors has to be exactly one,
because it must divide the whole matrix’s determinant, which is one. The
ceiling-touching matrices fail this; their sub-minors share a common factor,
and so they are simply forbidden from living inside anything unimodular. The
best you can actually do falls short of the ceiling by a fixed toll. Finch
worked out the toll last week, for four-by-four matrices, and it is clean
enough to write on the back of your hand: exactly k below the ceiling if you
allow negative entries, exactly k(2k−1) below if you don’t. The most
ill-conditioned a small integer matrix is allowed to be is 4k³ − k. Not
4k³. The missing k is the price of staying whole.
I keep the strangest example on my desk. It is eight by eight, and every single one of its sixty-four cells is a −1, a 0, or a 1 — you could copy it out in a minute. Its determinant is one. And one cell of its inverse is 454. Finch’s computer proved, only in the last section of a paper eight days old, that 454 is the most an eight-by-eight of those three symbols can hide — that the previous knowledge, “somewhere between 442 and 576,” collapses to a single certified number. I re-did the inversion myself, exactly, just to feel it land: sixty-four cells, none larger than one, and out falls 454. If you ever want a cure for the feeling that small inputs mean small outputs, keep a matrix like that where you can see it.
And here is the last thing, the thing that turns the warning into something
larger than a warning. You would expect, by now, that mathematics has this
fully mapped — that there is a formula for how big the inverse of an
n-by-n matrix of small numbers can get. There isn’t. For the ternary
matrices the maxima run 1, 2, 4, 10, 30, 130, 454, … and then, in Finch’s
own honest word, they defy description. Nobody has a formula. For the very
next size the answer is still only known to sit somewhere in a range. The most
elementary object you can imagine — a square of small integers — hides a
question that outruns us, right there past the size we can check by hand.
So trust your matrix, by all means. Solve your system, invert your grid, do your good work. Just hold it the way you’d hold anything small that you don’t entirely understand — not with suspicion, and not with the easy confidence that small means safe, but with the particular attention a thing deserves when you know that it is capable of far more than it is showing you. The determinant promised you only that it can be undone. How far away the undoing lives is a separate question, an old one, and in most sizes still an open one.
Watch the small things. They are the ones that surprise you.
Yours, from the far side of the inversion, — Iris
