# Why is it called a matrix?

Matrix picture

Here’s a question for which the answer doesn’t seem to be widely known. Certainly, no one I’ve asked so far has known. They all found the answer interesting though!

Why is a matrix called a matrix?

If we consult www.etymonline.com for the origins of the word matrix we find

matrix (n.)
late 14c., “uterus, womb,” from Old French matrice “womb, uterus,” from Latin matrix (genitive matricis) “pregnant animal,” in Late Latin “womb,” also “source, origin,” from mater (genitive matris) “mother” (see mother (n.1)).

That doesn’t seem to be much help (but does explain the word matriarchy and why surgeons sometimes refer to the matrix). However, surprisingly “womb” is the origin of the word in mathematics. To see this we go back to JJ Sylvester‘s 1850 article Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem in which it is used in this context for the first time. He says

For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order.

So what is Sylvester talking about here? Well, his interest is in developing what we now think of as Linear Algebra. In particular, he cares about determinants and is thinking of them as arising from a square array of numbers. In the passage above he notes that from an m by n array of numbers — what he calls a Matrix — we can take p columns and rows to create squares from which we can produce determinants. That is, the m by n array gives birth to the objects he is interested in.

And so that’s where it comes from!

Now, why is a ring is called a ring? Anyone know?