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I am curious about who first noticed that block matrices can be multiplied blockwise.

There is a section about matrices partitioned into submatrices that describes block matrix multiplication in "An introduction to the theory of canonical matrices" by Turnbull and Aitken, but there are earlier references that use block matrices and their multiplication and inverses, for example, Schur's dissertation "Über eine Klasse von Matrizen...". A natural guess for the origin of block matrices would be Jordan's "Traité des substitutions et des équations algebriques", but it seems from the formulas that Jordan does not employ block matrices and I don't read French, so I cannot say if Jordan mentions something in text.

Is there a reference earlier that Turnbull–Aitken that talks explicitly about multiplication of block matrices or was it a folklore simple thing?

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Matrix multiplication was defined by Cayley (1855, 1858), so anything earlier can only involve products of determinants, not of the matrices themselves. According to Cayley (1889),

The next later memoir on the theory of Matrices, so far as I am aware is that by Laguerre, “Sur le Calcul des Systèmes Lineaires,” Jour. Ec. Polyt. t. xxv. (1867), pp. 215–264. A “système lineaire” is what I called a matrix (...)

Lo and behold, Laguerre’s remarkable paper does introduce blockwise multiplication, pp. 255–256.

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In section 20 of the supplement to his "Theoria combinationis" Gauss proposed the solution of the normal equations in groups, which is perhaps not the kind of block multiplication that you are looking for, but in modern terms this method would be called a Block Gauss-Seidel method. Here is an English description.

Alternatively, here is a short note from 1858 by Zehfuss introducing the Kronecker product of two matrices.

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