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Evaluate the following determinant: $$\begin{vmatrix}
1 &2 & 3& \dots & n-1 &n \\
-1&0 & 3& \cdots &n-1 & n\\
-1& -2 & 0& \cdots & n-1 &n \\
\vdots &\vdots &\vdots & \cdots &\vdots & \vdots\\
-1& -2 & -3 & \cdots & 0 &n \\
-1& -2& -3 & \cdots & 1-n& 0
\end{vmatrix}$$

From every column \(C_k\,,k=2,3,\ldots,n\) of the determinant we substract \(k\)-times the \(C_1\) column: \begin{align*}
\left|{\begin{array}{rrrccc}
1 &2 & 3& \dots & n-1 &n \\ -1&0 & 3& \cdots &n-1 & n\\ -1& -2 & 0& \cdots & n-1 &n \\ \vdots &\vdots &\vdots & \ddots &\vdots & \vdots\\ -1& -2 & -3 & \cdots & 0 &n \\ -1& -2& -3 & \cdots & 1-n& 0\end{array}}\right| &\mathop{=\!=\!=\!=\!=\!=\!=\!=\!=}\limits^{\begin{subarray}{c}
{C_k\,\to \,C_k-kC_1} \\
{k\,=\,2,3,\ldots,n}
\end{subarray}}\left|{\begin{array}{rrrccc}
1 &0 & 0& \dots & 0 &0 \\
-1&2 & 0& \cdots &0 & 0\\
-1& 0 & 3& \cdots & 0 &0 \\
\vdots &\vdots &\vdots & \ddots &\vdots & \vdots\\
-1& 0 & 0 & \cdots & n-1 &0 \\
-1& 0& 0& \cdots & 0& n\end{array}}\right| \\
&=1\cdot\left|{\begin{array}{ccccc}
2 & 0& \cdots &0 & 0\\
0 & 3& \cdots & 0 &0 \\
\vdots &\vdots & \ddots &\vdots & \vdots\\
0 & 0 & \cdots & n-1 &0 \\
0& 0& \cdots & 0& n\end{array}}\right|\\
&=1\cdot2\cdot3\cdots(n-1)\,n\\
&=n!\,.
\end{align*}

Determinants were originally considered with no reference to the matrices. They were seen as a property instead of linear equation systems. If the system is non-zero, the determinant can 'determine' whether there is a unique solution for the system. Using this definition, the Nine Chapters on the Mathematical Arts from China, dating to around the 3rd century BC is the first use of determinants, ashttp://yourhomeworkhelp.org/do-my-math-homework/ writes.
European mathematicians considered two-by-two determinants around the end of the 16th century by Cardano and larger ones were explored by Leibniz(https://math.dartmouth.edu/archive/m8s00/public_html/handouts/matrices3/node7.html). Cramer worked on the theory as relative to sets of equation in 1750. Gauss applied the term to the discriminant of a quantic, and further explored notions of reciprocal (inverse) determinants and he came close to discovering the multiplication theorem. Binet, Cauchy, Jacobi, Sylvester and others went on to subsequently refine the theory.