Apparently the following matrix
$$A=\begin{bmatrix}
2 & 4
\\\ 1 & 2
\end{bmatrix}$$
is a composition of two linear transformations, $BC=A$.
The objective of this exercise is to decompose $A$ into $B,C\neq I_2$ such that $B$ and $C$ are either a rotation, scaling, projection, shear, or reflection.
It seems that all points end up on the line $y=\frac{1}{2}x$, so I tried making $C$ a projection onto the vector $\langle 2,1\rangle$, but then I cannot find a transformation $B$ that works.
Best Answer
Answer to the question before it was edited: for any non-singular matrix $S$ we can write $A= (AS) S^{-1}$ and we can take $B=AS, C=S^{-1}$.