Consider two matrices acting on points in $(0,1)^2$ in the real plane, $$h_s=\begin{pmatrix} e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix}.$$
and $$g_s=\begin{pmatrix} 1-e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix}.$$
How do you compose these transformations?
I tried to do $h_s \circ g_s$ but got stuck because I haven't dealt with composition of matrices before.
I have done function composition, so I am familiar with that concept.
Treat a point $(x,y)$ as a $2 \times 1$ vector $\begin{bmatrix} x \\ y \end{bmatrix}$. You may want to start with usual calculations for function compositions
\begin{align} g_s \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) &= \begin{pmatrix} 1-e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} (1-e^{-e^{s}})x \\ e^{-e^{-s}}y \end{pmatrix}. \tag{1} \label{my1} \\ (h_s \circ g_s) \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) &= h_s\left( g_s\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) \right) \\ &= h_s\left( \begin{pmatrix} (1-e^{-e^{s}})x \\ e^{-e^{-s}}y \end{pmatrix} \right) \\ &= \begin{pmatrix} e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix} \begin{pmatrix} (1-e^{-e^{s}})x \\ e^{-e^{-s}}y \end{pmatrix} \\ &= \begin{pmatrix} e^{-e^{s}} (1-e^{-e^{s}})x \\ e^{-2e^{-s}}y \end{pmatrix}. \end{align}
From this, it's clear that $h_s \circ g_s = \begin{pmatrix} e^{-e^{s}}(1-e^{-e^{s}}) & 0 \\ 0 & e^{-2e^{-s}} \end{pmatrix}$.
However, it's preferable to describe $g_s$ and $h_s$ in words. In this way, you see this directly without writing the vector. Note that in a matrix product, the row $\begin{pmatrix} 1 & 0 \end{pmatrix}$ on the left means to "pick the first row in the matrix/vector on its right-hand side". It's easy to see why the first row of the right-hand side in \eqref{my1} is $(1-e^{-e^s})x$. The second row follows similarly.
With this verbal description of matrix product in mind, you can find the matrix product $$ \begin{pmatrix} e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix} \begin{pmatrix} 1-e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix} $$ using the matrices associated with $h_s$ and $g_s$. Note that the question asks for $h_s \circ g_s$, so you have to put $$\text{matrix for } h_s \cdot \text{matrix for } g_s.$$ Note that the matrices in this question are diagonal matrices, so you can simply do an element-by-element multiplication on the diagonal.