Why are invertible matrices "one-to-one"?
I think the question is referring to the natural use of a matrix to map points to points by multiplication.
Suppose further that
Then multiplying both sides by
#p_1 = I p_1 = M^(-1)M p_1 = M^(-1)M p_2 = I p_2 = p_2#
#Mp_1 = Mp_2 => p_1 = p_2#
That is: multiplication by