# 矩阵迹与行列式问题

Let $A$ be an $n\times n$ matrix with real entries. Prove that $\det(A)=\frac{1}{n!}\det\left( \begin{matrix} \text{tr}(A)&1&0&0&\cdots&0\\\text{tr}(A^2)&\text{tr}(A)&2&0&\cdots&0\\\text{tr}(A^3)&\text{tr}(A^2)&\text{tr}(A)&3&\ddots&\vdots\\\vdots&\vdots&\ddots&\ddots&\ddots&0\\\text{tr}(A^{n-1})&\text{tr}(A^{n-2})&\cdots&\text{tr}(A^2)&\text{tr}(A)&n-1\\\text{tr}(A^n)&\text{tr}(A^{n-1})&\cdots&\cdots&\text{tr}(A^2)&\text{tr}(A)\end{matrix}\right)$.