Linear Algebra - Ch2 Determinants

Chapter 2   Determinants

2.1   Determinants of a Matrix

• Determinants Definition 

• 特殊型matrix的Determinants計算

• 轉置矩陣：轉置det(A)不變，det(A^T) = det(A)
• 三角矩陣：det(A)為對角線相乘
• Vandermonde matrix

• 基本矩陣的det(E)操作

• Determinants 性質 
• A is singular iff det(A) = 0  (用rref, elementary matrix的概念證)
• 有零列(行)或相同比例的列(行)則det(A) = 0
• det(A^-1) = 1 / det(A) 
• det(AB) = det(A)det(B)  (任何兩個方陣)
• det[ A  C ] = det(A)det(B)    
       [ O  B ]

2.2   Cramer's Rule

• Classical Adjoint and Inverse matrix

• det(adj(A))=(det(A))^n-1 (If A is any n x n non-singular matrix)
• proof : 
• A(adj(A)) = det(A) I
• det (A(adj(A))) = det (det(A) I)
  // 兩邊取det()，注意右側，det(K*I) = K^n
• det(A)det(adj(A)) = (det(A))^n
• det(adj(A)) = [det(A)]^(n - 1)

• Cramer's rule