Prove that the scalar 1 is the identity for scalar multiplication: 1A=A.?

Answer 1

How I would prove it is by thinking what multiplication is.

multiplication is a sort way to add.

like 2*5 = 5+5 = 2+2+2+2+2

or 1*2 = 1+1 and 1*10 = 1+1+1+1+1+1+1+1+1+1

so 1*(any number) would equal 1 added that many times

or (any number) it self...

Answer 2

An identity object in mathematics is something that does not change other subjects. Since 1 times x is always x, the x (A in your example) doesn't change. Therefore it is an identity object.