<s>[INST] Evaluation of a specific determinant. 
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.

$$A_2
\begin{pmatrix}  1&  1\\
  1&  2
\end{pmatrix};
A_3 = \begin{pmatrix}  1&  1&  1\\
  1&  2&  2\\
  1&  2&  3
\end{pmatrix};
A_4 = \begin{pmatrix}  1&  1&  1&  1\\
  1&  2&  2&  2\\
  1&  2&  3&  3\\
  1&  2&  3&  4
\end{pmatrix};
A_5 = \begin{pmatrix}  1&  1&  1&  1&  1\\
  1&  2&  2&  2&  2\\
  1&  2&  3&  3&  3\\
  1&  2&  3&  4&  4\\
  1&  2&  3&  4&  5
\end{pmatrix}$$
$$A_6 = \begin{pmatrix}  1&  1&  1&  1&  1&  1\\
  1&  2&  2&  2&  2&  2\\
  1&  2&  3&  3&  3&  3\\
  1&  2&  3&  4&  4&  4\\
  1&  2&  3&  4&  5&  5\\
  1&  2&  3&  4&  5&  6
\end{pmatrix};
A_7 = \begin{pmatrix}  1&  1&  1&  1&  1&  1&  1\\
  1&  2&  2&  2&  2&  2&  2\\
  1&  2&  3&  3&  3&  3&  3\\
  1&  2&  3&  4&  4&  4&  4\\
  1&  2&  3&  4&  5&  5&  5\\
  1&  2&  3&  4&  5&  6&  6\\
  1&  2&  3&  4&  5&  6&  7
\end{pmatrix}
$$
 [/INST] $$A_2 = \begin{bmatrix}1 & 1\\ 1 & 2 \end{bmatrix} = \begin{bmatrix}1 & 0\\ 1 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}$$
$$A_3 = \begin{bmatrix} 1 & 1 & 1\\ 1 & 2 & 2\\1 & 2 & 3 \end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\end{bmatrix}$$
$$A_4 = \begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 2 & 2 & 2\\1 & 2 & 3 & 3\\ 1 & 2 & 3 & 4\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\end{bmatrix}$$
Can you see the pattern here? Prove this is the case in general.
Use this, along with the fact that $\det(XY) = \det(X) \cdot \det(Y)$ and determinant of a triangular matrix is the product of the diagonal entries to conclude the answer.
EDIT
Note that $$A_{ij} = \min(i,j) = \sum_{k=1}^{\min(i,j)} 1 = \sum_{k=1}^{n} \mathbb{I}_{k \leq i}\mathbb{I}_{k \leq j} = \sum_{k=1}^n B_{ik} C_{kj}$$
where $B_{ik} = \begin{cases} 1 & k \leq i\\ 0 & \text{else}\end{cases}$ and $C_{kj} = \begin{cases} 1 & k \leq j\\ 0 & \text{else}\end{cases}$.
This should enable you to get the decomposition.
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