It does not require commutativity of multiplication to see that 5 * 3 = 5 + 5 + 5. This property has nothing whatsoever to do with commutativity, only the distributivity laws for multiplication over addition, the definition of the symbols 5 and 3 as sums of 1’s (or some equivalent lemma), and the multiplicative identity laws.

That is:

5 * 3

= 5 * ((1 + 1) + 1) — One possible definition of 3

= 5 * (1 + 1) + 5 * 1 — Distributivity of multiplication over addition (on the left)

= (5 * 1 + 5 * 1) + 5 * 1 — Distributivity again

= (5 + 5 * 1) + 5 * 1 — Multiplicative identity (on the right)

= (5 + 5) + 5 * 1 — Multiplicative identity

= (5 + 5) + 5 — Multiplicative identity

This identity holds in matrix rings just as well as it holds in the integers.

Similarly:

5 * 3

= ((((1 + 1) + 1) + 1) + 1) * 3 — Definition of 5

= ((((1 + 1) + 1) + 1) * 3 + 1 * 3) — Distributivity (on the right)

= ((((1 + 1) + 1) * 3 + 1 * 3) + 1 * 3) — Distributivity

= ((((1 + 1) * 3 + 1 * 3) + 1 * 3) + 1 * 3) — Distributivity

= ((((1 * 3 + 1 * 3) + 1 * 3) + 1 * 3) + 1 * 3) — Distributivity

= ((((3 + 1 * 3) + 1 * 3) + 1 * 3) + 1 * 3) — Mult. Identity (on the left)

= ((((3 + 3) + 1 * 3) + 1 * 3) + 1 * 3) — Mult. Identity

= ((((3 + 3) + 3) + 1 * 3) + 1 * 3) — Mult. Identity

= ((((3 + 3) + 3) + 3) + 1 * 3) — Mult. Identity

= ((((3 + 3) + 3) + 3) + 3) — Mult. Identity

I wouldn’t expect to need to write this out, but I don’t understand how so many people are missing this.

As for the points about arrays, the use of × as “by” there isn’t really multiplication. After all, you couldn’t say “15 array” in any place where you might otherwise say “5 × 3 array”.