Let's suppose for a matrix $A$, the adjoint is $adjA$, the determinant is $detA$ and the inverse is $A^{-1}$. If I am now considering a matric $kA$, what can I say about the relation between $A$ and $kA$, between $adj(A)$ and $adj(kA)$, between $det(A)$ and $det(kA)$, between $A^{-1}$ and $(kA)^{-1}$
If the answer depends on the order of the matrix, which I believe it should according to my preliminary analysis, consider only for $2\times2$ or $3\times3$ orders. Also, you can change the notation if you want to...