2024年考研数学三第6题
📝 题目
设 $A$ 为 3 阶矩阵，$P=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right)$ ．若 $P^T A P^2=\left(\begin{array}{ccc}a+2 c & 0 & c \\ 0 & b & 0 \\ 2 c & 0 & c\end{array}\right)$ ，则 $A=$（ ）_
💡 答案解析
答案: C
解析:
$P^{T} A P^{2}=\left(\begin{array}{ccc}a+2 c & 0 & c \ 0 & b & 0 \ 2 c & 0 & c\end{array}\right)$ ，
则 $A=\left(P^{T}\right)^{-1}\left(\begin{array}{ccc}a+2 c & 0 & c \ 0 & b & 0 \ 2 c & 0 & c\end{array}\right)\left(P^{2}\right)^{-1}=\left(\begin{array}{ccc}1 & 0 & -1 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right)\left(\begin{array}{ccc}a+2 c & 0 & c \ 0 & b & 0 \ 2 c & 0 & c\end{array}\right)\left(\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ -1 & 0 & 1\end{array}\right)^{2}=\left(\begin{array}{ccc}a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c\end{array}\right)$
故选 C。