Suppose $A=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i \end{pmatrix}$ and $\det(A) = 1$. What does that tell us about the $\text{RREF}$ of the matrix
$$B=\begin{pmatrix}a&b&c&\lambda_1\\d&e&f&\lambda_2\\g&h&i&\lambda_3 \end{pmatrix}$$
In particular, is the $\text{RREF}$ of $B$ such that the non-$\lambda$ columns (essentially the matrix $A$ in $B$, if that makes sense) reduce to the identity matrix?
$$B=\begin{pmatrix}1&0&0&\lambda_1'\\0&1&0&\lambda_2'\\0&0&1&\lambda_3' \end{pmatrix}$$
For some new where $\lambda'$s are scalar multiples of $\lambda$'s.