It is proved that: 1. (1) M ƒ λ is a substitution closed full AFL closed also under doubling ( D( L) = { xx | x ϵ L}). 2. (2) M ƒ = M ƒ λ = M ac ƒ = M ac ƒ λ = M left ƒ = M left ƒ λ = PM ƒ = PM ƒ λ! 3. (3) LM ƒ ⊂ M ƒ, strict inclusion. (The families M - M left λ are as in Salomaa's book [18], LM is the family or simple matrix languages of Ibarra [11] and PM, PM λ are the families of λ-free, respectively, arbitrary, parallel matrix languages [4]. For an arbitrary family of languages, L we denote by L ƒ the family of finite index languages in L , the index being considered with respect to the class of grammars generating languages in L .) 4. (4) The emptiness and the finiteness problems for M ƒ are algorithmically solvable, whereas many other decision problems about M ƒ are unsolvable. 5. (5) The composition does not increase the generative capacity of (context-free), arbitrary or of finite index, matrix grammars. A conjecture with many significant implications is formulated in the last part of the paper.