考研线性代数：行列式部分初级专项练习题（2024 年）

题目 09

$$D=\left|\begin{array}{lll}{a}^2& (a+2{)}^2& (a+4{)}^2\\ b^2& (b+2{)}^2& (b+4{)}^2\\ c^2& (c+2{)}^2& (c+4{)}^2\end{array}\right|=?$$

解析 09

$$D=\left|\begin{array}{lll}{a}^2& (a+2{)}^2& (a+4{)}^2\\ b^2& (b+2{)}^2& (b+4{)}^2\\ c^2& (c+2{)}^2& (c+4{)}^2\end{array}\right|\Rightarrow$$

$$D=\left|\begin{array}{lll}{a}^2& a^2+4+4a& a^2+16+8a\\ b^2& b^2+4+4b& b^2+16+8b\\ c^2& c^2+4+4c& c^2+16+8c\end{array}\right|\Rightarrow$$

$$D=\left|\begin{array}{lll}{a}^2& 4+4a& 8a+16\\ b^2& 4+4b& 8b+16\\ c^2& 4+4c& 8c+16\end{array}\right|\Rightarrow$$

$$D=4\times 8\left|\begin{array}{lll}{a}^2& a+1& a+2\\ b^2& b+1& b+2\\ c^2& c+1& c+2\end{array}\right|\Rightarrow$$

$$D=32\left|\begin{array}{lll}{a}^2& a+1& 1\\ b^2& b+1& 1\\ c^2& c+1& 1\end{array}\right|\Rightarrow$$

$$D=32\left|\begin{array}{lll}{a}^2& a& 1\\ b^2& b& 1\\ c^2& c& 1\end{array}\right|\Rightarrow$$

$$D=32\times (-1{)}^{2+1}\Rightarrow$$

$$\left|\begin{array}{lll}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|\Rightarrow$$

$$D=-32\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|\Rightarrow$$

$$D=-32(b-a)(c-a)(c-b).$$