This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative.

In the middle ages the metal chiefly employed in the manufacture of matrices was bronze.

Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created.

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In the later centuries also, particularly in the 14th century, they were set in seal matrices and finger rings.

In the British Museum are the bronze matrices of seals of ZEthilwald, bishop of Dunwich, about Boo; of lElfric, alderman of Hampshire, about 985; and the finely carved ivory double matrix of Godwin the thane (on the obverse) and of the nun Godcythe (on the reverse), of the beginning of the 11th century.

As already stated, the matrices of ancient Babylonian and Assyrian seals, usually cut on precious stones, are in cylinder form.

It has usually been the custom to break up or deface the matrices of official seals when they have ceased to be valid, as, for example, at the commencement of a new reign.

But there are examples of elaborate matrices composed of several pieces, from the impressions of which the seal was built up in an ingenious fashion, both obverse and reverse being carved in hollow work, through which figures and subjects impressed on an inner layer of wax are to be seen.

Examples of seals, both matrices and impressions, are found among the antiquities of Egypt, Babylonia and Assyria.

One of the oldest matrices is an intaglio in rock crystal, now preserved at Aix-la-Chapelle, bearing a portrait head of Lothair II., king of Lorraine (A.D.