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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. $$ \det(A) = \frac 1{n!} \left| \begin{array}{cccccc}\operatorname{tr}(A) & 1 & 0 & \cdots & \cdots & 0 \\ \operatorname{tr}(A^2) & \operatorname{tr}(A) & 2 & 0 & \cdots & 0 \\ \operatorname{tr}(A^3) & \operatorname{tr}(A^2) & \operatorname{tr}(A) & 3 & & \vdots \\ \vdots & & & & & n-1 \\ \operatorname{tr}(A^n) & \operatorname{tr}(A^{n-1}) & \operatorname{tr}(A^{n-2}) & \cdots & \cdots & \operatorname{tr}(A) \end{array}\right|$$

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@DietrichBurde yes. –  user117432 Jan 1 at 14:32
    
why is this helpful for you? What is the background for this? What have you tried on your part? do you immediately see this for a $2\times 2$ matrix? –  Praphulla Koushik Jan 1 at 14:34
    
This looks like a result of the Leverrier-Faddejev algorithm. At the same time, the matrix is part of the system of Newton identities relating the power sums of the eigenvalues to the coefficients of the characteristic polynomial. –  LutzL Jan 1 at 14:34

2 Answers 2

up vote 12 down vote accepted

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}

or

$$-\begin{bmatrix} s_1\\s_2\\s_3\\\vdots\\s_{n-1}\\s_n \end{bmatrix} = \begin{bmatrix} 1&0&0&\dots&0&0\\ s_1&2&0&\dots&0&0\\ s_2&s_1&3&&0&0\\ \vdots&\vdots&&&&\vdots\\ s_{n-2}&s_{n-3}&s_{n-4}&\dots&n-1&0\\ s_{n-1}&s_{n-2}&s_{n-3}&\dots&s_1&n \end{bmatrix} \begin{bmatrix} c_1\\c_2\\c_3\\\vdots\\c_{n-1}\\c_n \end{bmatrix}$$

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.

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Determinants and traces are invariant and everything is polynomial, so it is enough to check it for diagonal matrices.

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