In linear regression the OLS solution is given by:
$$ \hat{\beta} = (X^TX)^{-1}X^TY $$
I want to show that if you scale the $i$th predictor variable by a constant, then the corresponding $i$th coefficient scales down.
In matrix notation I want to show:
If we scale the $i$th column
$$ X^{*}_{(i)} = a \cdot X_{(i)} $$
where $X_{(i)}$ denotes the $i$th column, then:
$$\hat{\beta}^*_i = \frac{\hat{\beta}_i}{a}$$
