Question: I am curious how I can make something similar to this

[![enter image description here](https://i.stack.imgur.com/tdmVu.png)](https://i.stack.imgur.com/tdmVu.png)

Answer: 
I have just chose the way to recreate your picture with the clone of Times New Roman.

```
\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{newtxtext,newtxmath}

\begin{document}
\[\exp(At_y)=\begin{pmatrix}
\exp(t \begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix}) & \begin{matrix}
0 \\
0 \\
0
\end{matrix} \\
\begin{matrix}
0 & 0 & 0
\end{matrix}  & \exp(t(1)) 
\end{pmatrix}\]
\end{document}

```

[![enter image description here](https://i.stack.imgur.com/zrrjM.png)](https://i.stack.imgur.com/zrrjM.png)

Using a `smallmatrix` you will have:

```
\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{newtxtext,newtxmath}

\begin{document}
\[\exp(At_y)=\begin{pmatrix}
\exp(t \Bigl(\begin{smallmatrix}
1 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{smallmatrix}\Bigr)) & \begin{matrix}
0 \\
0 \\
0
\end{matrix} \\
\begin{matrix}
0 & 0 & 0
\end{matrix}  & \exp(t(1)) 
\end{pmatrix}\]
\end{document}

```

[![](https://i.stack.imgur.com/6U7k9.png)](https://i.stack.imgur.com/6U7k9.png)

A last possibility with the `medsize` environment from `nccmath`:

```
\usepackage{nccmath} 
\newenvironment{medpmatrix}{\medsize\begin{pmatrix}}{\end{pmatrix}\endmedsize}
................
\[ \exp(At_y)=\begin{pmatrix}
\exp(t \begin{medpmatrix}
1 & 1 & 0\\[-0.4ex]
0 & 1 & 0\\[-0.4ex]
0 & 0 & 1
\end{medpmatrix}\mathrlap{)} & \begin{matrix}
0 \\
0 \\
0
\end{matrix} \\
\begin{matrix}
0 & 0 & 0
\end{matrix} & \exp(t(1))
\end{pmatrix} \]

```

[![enter image description here](https://i.stack.imgur.com/i66IO.png)](https://i.stack.imgur.com/i66IO.png)
My proposal. I prefer brackets for the matrices, because parentheses are already overused.

The trick is to nest matrices and to add a small vertical space between the two rows.

```latex
\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[
\exp(tA_j)=\begin{bmatrix}
  \exp\left( t \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\right) &
  \begin{matrix} 0 \\ 0 \\ 0 \end{matrix}
\\ \noalign{\vspace{0.5ex}}
0\qquad 0\qquad 0 & \exp\bigl(t(1)\bigr)
\end{bmatrix}
\]

\[
\exp(tA_j)=\begin{pmatrix}
  \exp\left( t \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\right) &
  \begin{matrix} 0 \\ 0 \\ 0 \end{matrix}
\\ \noalign{\vspace{0.5ex}}
0\qquad 0\qquad 0 & \exp\bigl(t(1)\bigr)
\end{pmatrix}
\]

\end{document}

```

[![enter image description here](https://i.stack.imgur.com/wuJvP.png)](https://i.stack.imgur.com/wuJvP.png)