Question

What is the multiplication identity property?

Answer 1

For a set of elements, `S` and an operation (called multiplication and indicated by the symbol `xx` in this explanation).

If for all `x` which are members of `S` if there is one element `phi` of `S` for which

`phi xx x = x` and `x xx phi = x`
(for all `x epsilon S`)

Then `phi` is called the multiplicative identity
and
`phi xx x = x` is called the multiplicative identity property .

For Integers, Rational Numbers, Real Numbers and Complex Numbers the multiplicative identity is `1`.

That is
(any number) `xx 1 =` (the same number).

For matrices the multiplicative identity is the Identity Matrix

For a more complex set and an operation that we might not normally think of as "multiplication",
the multiplicative identity `phi` might be something quite different provided it satisfies the multiplicative identity property for that set and operation.