English
For 2×2 matrices A and B over a nonunital semiring, the entries of AB are given by the standard bilinear sums; in particular, the (0,0) entry equals A00 B00 + A01 B10, etc.
Русский
Для матриц 2×2 A и B над полукольцом без единицы элементы произведения AB выражаются через стандартные билинейные суммы; в частности (0,0)-й элемент равен A00 B00 + A01 B10 и т.д.
LaTeX
$$$(A B)_{00} = A_{00} B_{00} + A_{01} B_{10} \land (A B)_{01} = A_{00} B_{01} + A_{01} B_{11} \land (A B)_{10} = A_{10} B_{00} + A_{11} B_{10} \land (A B)_{11} = A_{10} B_{01} + A_{11} B_{11}$$$
Lean4
@[simp]
protected theorem mul_zero [Fintype n] (M : Matrix m n α) : M * (0 : Matrix n o α) = 0 :=
by
ext
apply dotProduct_zero
