# Why are invertible matrices "one-to-one"?

Dec 20, 2015

See explanation...

#### Explanation:

I think the question is referring to the natural use of a matrix to map points to points by multiplication.

Suppose $M$ is an invertible matrix with inverse ${M}^{- 1}$

Suppose further that $M {p}_{1} = M {p}_{2}$ for some points ${p}_{1}$ and ${p}_{2}$.

Then multiplying both sides by ${M}^{- 1}$ we find:

${p}_{1} = I {p}_{1} = {M}^{- 1} M {p}_{1} = {M}^{- 1} M {p}_{2} = I {p}_{2} = {p}_{2}$

So:

$M {p}_{1} = M {p}_{2} \implies {p}_{1} = {p}_{2}$

That is: multiplication by $M$ is one-to-one.