现代知识点小结.pptVIP

* Theorem 2.1. 4 ( P90) : Let A be an n×n matrix : (1) If A has a row or column consisting entirely of zeros, then det(A)=0. (1+n) + (1+n-1) + …… + (1+2)+(1+1) = n + (n+ n-1+……+ 2+1 ) (2) If A has two indentical rows or columns, then det(A)=0. Proof : (1). n=2 , the result holds. = 0 It can be expaned along the third row. (2) by induction on n . §2 Properties of Determinants ( P91 ) 1. Lemma 2.1.1 : When i ≠ j , then : When i = j , then : When j ≠ k , then : Proof : ……….. §2 Properties of Determinants ( P91 ) 1. Lemma 2.1.1 : When i ≠ j , then : When i = j , then : When j ≠ k , then : Proof : ……….. Solution: Remark : thinking following question A isn?t invertible ? A is singular. AX=0 has infinity solutions. (c) A isn?t row equivalent to I . (d) There does?t exist B such AB=BA=I (e) Det (A ) = 0 A simple summary: The Adjoint of A is : §3 Additional Topics ( P98 ) Proof : Remark : What will happen when det (A) =0 ? Rewrite : Thus : P99 Example 1 , 2 ( P99-100 ) (一）AX=0 （二）AX=b It has a unique solution : Summary : Two methods to solve linear equations: 1. Elementary row transformation 2. Determinant It has a trivial solution : Det(A)≠0 It has nontrivial solutions : Det(A)=0 or m n in reduced row echelon form of A ① When Det(A)≠0 (AX=0 is called homogeneous systems) (AX=b is called nonhomogeneous systems) *