How to prove that $\det(A)$ can be expressed as a $n \times n$ determinant with entries $\operatorname{tr}(A^k)$ Let $A$ be an $n\times n$ non-singular matrix with real entries. How can I prove the following equation? Any references would be helpful. 
$$ \det(A) = \frac 1{n!} \left| \begin{array}{cccccc}\operatorname{tr}(A) & 1 & 0 & \cdots &  \cdots & 0 \\ \operatorname{tr}(A^2) & \operatorname{tr}(A) & 2 & 0 & \cdots & 0 \\ \operatorname{tr}(A^3) & \operatorname{tr}(A^2) & \operatorname{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \operatorname{tr}(A^n) & \operatorname{tr}(A^{n-1}) &  \operatorname{tr}(A^{n-2}) & \cdots & \cdots & \operatorname{tr}(A)            \end{array}\right|$$
 A: Determinants and traces are invariant and everything is polynomial, so it is enough to check it for diagonal matrices.
A: The Newton identities relating power sums of eigenvalues $s_k=tr(A^k)$ to the coefficients of the characteristic polynomial $\chi_A(t)=t^n+c_1t^{n-1}+\dots+c_{n-1}t+c_n$ with $c_n=(-1)^n\det(A)$ read as
\begin{align}
  s_1 &= -c_1,\\
  s_2 &= -c_1 s_1 - 2 c_2,\\
  s_3 &= -c_1 s_2 - c_2 s_1 - 3 c_3,\\
  s_4 &= -c_1 s_3 - c_2 s_2 - c_3 s_1 - 4 c_4, \\
         & {} \  \  \vdots\\
  s_n &= -c_1 s_{n-1}-\dots-c_{n-1} s_1 - n c_n
\end{align}
which can be written as a matrix-vector system 
$$-\begin{bmatrix}
s_1\\s_2\\s_3\\\vdots\\s_{n-1}\\s_n
\end{bmatrix}
=
\begin{bmatrix}
1&0&0&\dots&0&0\\
s_1&2&0&\dots&0&0\\
s_2&s_1&3&&0&0\\
\vdots&\vdots&&&&\vdots\\
s_{n-2}&s_{n-3}&s_{n-4}&\dots&n-1&0\\
s_{n-1}&s_{n-2}&s_{n-3}&\dots&s_1&n
\end{bmatrix}
\begin{bmatrix}
c_1\\c_2\\c_3\\\vdots\\c_{n-1}\\c_n
\end{bmatrix}$$
or
$$\begin{bmatrix}
1\\0\\0\\0\\\vdots\\0\\0
\end{bmatrix}
=
\begin{bmatrix}
1&0&0&0&\dots&0&0\\
s_1&1&0&0&\dots&0&0\\
s_2&s_1&2&0&\dots&0&0\\
s_3&s_2&s_1&3&&0&0\\
\vdots&\vdots&\vdots&&&&\vdots\\
s_{n-1}&s_{n-2}&s_{n-3}&s_{n-4}&\dots&n-1&0\\
s_n&s_{n-1}&s_{n-2}&s_{n-3}&\dots&s_1&n
\end{bmatrix}
\begin{bmatrix}
1\\c_1\\c_2\\c_3\\\vdots\\c_{n-1}\\c_n
\end{bmatrix}$$
Now apply Cramers rule to the computation of $c_n$ to obtain the stated formula.

Note that the solution of this triangular system constitutes the computational core of the Leverrier-Faddejev algorithm for the (mostly) division free computation of the characteristic polynomial of a matrix.
