= Solution
Rescaling the first basis vector gives the <change-of-basis matrix> $P=\operatorname{diag}(k,1)$, invertible because $k\ne0$. Direct multiplication gives
$$
\boxed{P^{-1}\begin{pmatrix}a&c\\b&d\end{pmatrix}P
=\begin{pmatrix}a&c/k\\bk&d\end{pmatrix}.}
$$
Thus the matrices are related by <matrix similarity>. In terms of basis images, the coefficient along the rescaled first basis vector is divided by $k$, while the second component of its image is multiplied by $k$.
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