{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-determinant"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-determinant-01","title":{"zh-Hant":"二階行列式：乘積公式、縮放公式與實數域可逆判準","en":"2×2 determinants: the product rule, the scaling rule and the invertibility criterion over ℝ"},"summary":{"zh-Hant":"對所有 2×2 實矩陣證明 det(AB)=det(A)det(B) 與 det(tM)=t²det(M)，並證明 M 在實數域可逆當且僅當 det(M)≠0。以 A=[[2,1],[−1,3]]、B=[[1,−2],[4,0]] 核對乘積公式並顯示 AB≠BA；以 t=−3 核對縮放公式；求 A⁻¹，並說明 A 雖是整數矩陣且 det(A)=7≠0，卻沒有整數反矩陣；以 C=[[1,2],[3,4]]（det=−2）說明非零行列式不必為正。","en":"For all 2×2 real matrices prove det(AB)=det(A)det(B) and det(tM)=t²det(M), and prove that M is invertible over ℝ exactly when det(M)≠0. With A=[[2,1],[−1,3]] and B=[[1,−2],[4,0]], check the product rule and show AB≠BA; check the scaling rule with t=−3; find A⁻¹ and explain why A, although an integer matrix with det(A)=7≠0, has no integer inverse; use C=[[1,2],[3,4]] (det=−2) to show that a nonzero determinant need not be positive."},"mko_ids":["mko-determinant"]}],"title":"概念推導導讀"}