I know how to do part a but b is giving me some trouble.
I know that the composition is $M\circ L=[M][L]$ but the matrices I get for $L$ and $M$ aren't compatible with multiplication.
For matrix L I get:
$$\begin{bmatrix}1&-2&-4\\2&1&7\end{bmatrix}$$
And by using the rotation standard matrix at $-\frac{\pi}{2}$ I get $M$:
$$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$
I know that $L$ is correct but these two matrixes aren't compatible with multiplication ($L$ is $2\times3$, $M$ is $2\times2$). Have I missed something with matrix $M$?
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Best Answer
The matrices are in fact compatible with multiplication. $[M]$ is a $2\times 2$ matrix and $[L]$ is a $2\times 3$ matrix, so you can multiply them: $$ [M][L] = \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \begin{bmatrix} 1 & -2 & -4\\ 2 & 1 & 7 \end{bmatrix} = \begin{bmatrix} 2 & 1 & 7\\ -1 & 2 & 4 \end{bmatrix}. $$