哈尔滨工业大学 2023年高等代数第0题

4．（1）$A=\left(\begin{array}{cccc}a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n}\end{array}\right), \quad F=\left(\begin{array}{cccc}1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & & \vdots \\ 1 & 1 & \cdots & 1\end{array}\right)$ ． 证 $|A+x F|=|A|+\sum_{i=1}^{n} \sum_{j=1}^{n} A_{i j}$ ． （2）$\left|\begin{array}{ccccc}1 & a_{12}-a_{11} & a_{13}-a_{12} & \cdots & a_{1 n}-a_{1, n-1} \\ 1 & a_{22}-a_{21} & a_{23}-a_{22} & \cdots & a_{2 n}-a_{2, n-1} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & a_{n 2}-a_{n 1} & a_{n 3}-a_{n 2} & \cdots & a_{n n}-a_{n, n-1}\end{array}\right|=\sum_{i=1}^{n} \sum_{j=1}^{n} A_{i j}$ ．