diff --git a/.translate/state/career.md.yml b/.translate/state/career.md.yml
index f05545de..50dc92f9 100644
--- a/.translate/state/career.md.yml
+++ b/.translate/state/career.md.yml
@@ -1,5 +1,5 @@
-source-sha: a4fbd8653600ac99163bfd5d4f7d98c1e613042e
-synced-at: "2026-07-26"
+source-sha: 0d8ef6103a5a1a5a340dc3bbddb8899a07645b09
+synced-at: "2026-10-22"
 model: claude-sonnet-5
 mode: UPDATE
 section-count: 4
diff --git a/.translate/state/harrison_kreps.md.yml b/.translate/state/harrison_kreps.md.yml
index 3a6e0404..9266ef65 100644
--- a/.translate/state/harrison_kreps.md.yml
+++ b/.translate/state/harrison_kreps.md.yml
@@ -1,5 +1,5 @@
-source-sha: 01e6f828d55d38d1ec82684b24fbb4e2a943de30
-synced-at: "2026-08-05"
+source-sha: 15d4f054f7663e4e6d9745d25636357526da8514
+synced-at: "2026-10-22"
 model: claude-sonnet-5
 mode: UPDATE
 section-count: 4
diff --git a/.translate/state/jv.md.yml b/.translate/state/jv.md.yml
index fbfe852e..22d7070c 100644
--- a/.translate/state/jv.md.yml
+++ b/.translate/state/jv.md.yml
@@ -1,5 +1,5 @@
-source-sha: a4fbd8653600ac99163bfd5d4f7d98c1e613042e
-synced-at: "2026-07-26"
+source-sha: 0d8ef6103a5a1a5a340dc3bbddb8899a07645b09
+synced-at: "2026-10-22"
 model: claude-sonnet-5
 mode: UPDATE
 section-count: 5
diff --git a/.translate/state/likelihood_bayes.md.yml b/.translate/state/likelihood_bayes.md.yml
index 74fd0fd3..96be5cd3 100644
--- a/.translate/state/likelihood_bayes.md.yml
+++ b/.translate/state/likelihood_bayes.md.yml
@@ -1,5 +1,5 @@
-source-sha: 07975d502f57b29b4aa331a71c0afa6809dee369
-synced-at: "2026-08-19"
+source-sha: 06c784f5e5939bd1379adeb2289488d84dfa7c8d
+synced-at: "2026-10-22"
 model: claude-sonnet-5
 mode: UPDATE
 section-count: 8
diff --git a/.translate/state/mccall_q.md.yml b/.translate/state/mccall_q.md.yml
index 19de1e97..b95228b0 100644
--- a/.translate/state/mccall_q.md.yml
+++ b/.translate/state/mccall_q.md.yml
@@ -1,5 +1,5 @@
-source-sha: 46cf999fa492255058c67e132b99735a7cb5fbaa
-synced-at: "2026-08-05"
+source-sha: 0d8ef6103a5a1a5a340dc3bbddb8899a07645b09
+synced-at: "2026-10-22"
 model: claude-sonnet-5
 mode: UPDATE
 section-count: 7
diff --git a/.translate/state/odu.md.yml b/.translate/state/odu.md.yml
index 5bf4f446..1d5306ac 100644
--- a/.translate/state/odu.md.yml
+++ b/.translate/state/odu.md.yml
@@ -1,5 +1,5 @@
-source-sha: 78030a3a27f6527046675bcd8a8d27995ca25af6
-synced-at: "2026-07-18"
+source-sha: 0d8ef6103a5a1a5a340dc3bbddb8899a07645b09
+synced-at: "2026-10-22"
 model: claude-sonnet-5
 mode: RESYNC
 section-count: 12
diff --git a/lectures/harrison_kreps.md b/lectures/harrison_kreps.md
index 39a5dfb2..cf701b4f 100644
--- a/lectures/harrison_kreps.md
+++ b/lectures/harrison_kreps.md
@@ -292,11 +292,14 @@ def price_single_beliefs(transition, dividend_payoff, β=.75):
     """
     求解单一信念的函数
     """
-    # 首先计算逆矩阵部分
-    imbq_inv = la.inv(np.eye(transition.shape[0]) - β * transition)
+    # 首先计算 (I - βQ)
+    A = np.eye(transition.shape[0]) - β * transition
 
-    # 接下来计算价格
-    prices = β * imbq_inv @ transition @ dividend_payoff
+    # 接下来计算 βQd
+    b = β * (transition @ dividend_payoff)
+
+    # 求解线性方程组
+    prices = la.solve(A, b)
 
     return prices
 ```