Just over 150 years ago, the matrices had their importance detected and emerged from the shadow of the determinants. The first to give them a name seems to have been Cauchy, 1826: tableau (= table).
The matrix name only came with James Joseph Sylvester, 1850. His friend Cayley, with his famous Memoir on the Theory of Matrices, 1858, divulged this name and began to demonstrate its usefulness.
He used the colloquial meaning of the word matrix, namely: place where something is generated or created. In fact, I saw them as "… A rectangular block of terms… which is not a determinant, but it is like a matrix from which we can form various systems of determinants by setting a number p and choosing p rows and p columns at will…" (article published in Philosophical Magazine 1850, pages 363-370).
Note that Sylvester still saw the matrices as a mere ingredient of the determinants. Only with Cayley do they come to have a life of their own and gradually begin to supplant the determinants in importance.