diff --git a/.translate/state/numpy.md.yml b/.translate/state/numpy.md.yml
index 94dc5d4..4cdee99 100644
--- a/.translate/state/numpy.md.yml
+++ b/.translate/state/numpy.md.yml
@@ -1,6 +1,6 @@
-source-sha: d35eb831da23a763f43d403e96f20d43529cae53
-synced-at: "2026-08-12"
+source-sha: ed3025d7f2b3710b4bfd622b9726999bbe5c78df
+synced-at: "2026-11-02"
 model: claude-sonnet-5
 mode: UPDATE
 section-count: 8
-tool-version: 0.25.0
+tool-version: 0.30.0
diff --git a/lectures/numpy.md b/lectures/numpy.md
index ad29a4a..1dd57bd 100644
--- a/lectures/numpy.md
+++ b/lectures/numpy.md
@@ -454,9 +454,6 @@ A * B
 ```{index} single: NumPy; Matrix Multiplication
 ```
 
-```{index} single: NumPy; Matrix Multiplication
-```
-
 我们使用 `@` 符号进行矩阵乘法，如下所示：
 
 ```{code-cell} python3