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Date: Sun, 18 Jan 2026 17:47:46 +0800
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---
 一些有趣的线代题目.md | 38 ++++++++++++++++++++++++++++++++++
 1 file changed, 38 insertions(+)

diff --git a/一些有趣的线代题目.md b/一些有趣的线代题目.md
index 0814fbb..4619d59 100644
--- a/一些有趣的线代题目.md
+++ b/一些有趣的线代题目.md
@@ -50,4 +50,42 @@ $$
 $$
 $$\text{rank}(A - E) =  k$$
 
+证毕
+设 $A, B$ 是 3 阶矩阵，$AB = 2A - B$，如果 $\lambda_1, \lambda_2, \lambda_3$ 是 $A$ 的 3 个不同特征值。证明：
+
+(1) $AB = BA$；
+
+(2) 存在可逆矩阵 $P$，使得 $P^{-1}AP$ 与 $P^{-1}BP$ 均为对角矩阵。
+
+**证明：**
+
+(1) ∵ $AB = 2A - B$
+
+$$
+\therefore (A - 2E)(B + E) = -2E
+$$
+
+$$
+\therefore (A - 2E)(B + E) = (B + E)(A - 2E)
+$$
+
+$$
+\therefore AB = BA
+$$
+
+(2) 设 $P^{-1}AP = \Lambda$，$\Lambda$ 为对角矩阵  
+则 $P^{-1}APBP = P^{-1}BAPP$
+
+$$
+\therefore P^{-1}APP^{-1}BP = P^{-1}BPP^{-1}APP
+$$
+
+设 $P^{-1}BP = N_2$
+
+$$
+\therefore \Lambda_1\Lambda_2 = \Lambda_2\Lambda_1
+$$
+
+与对角矩阵可交换的矩阵必为对角矩阵  
+
 证毕
\ No newline at end of file