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Chapter 2 Determinant

Chapter 2 Determinant. Chapter 2 行列式 Determinant. § 2-1 二階行列式. § 2-2 高階行列式. Define: A=nxn, A 之 determinant, det(A), 可遞迴定義如下：. (1) If n=1,. (2) If n=2,. submatrix. Property:. If: By induction on n: “induction” 歸納法”. (2) 假設 n=k 成立，接下來要証明 n=k+1 也成立：. consider n=k+1:. Let.

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Chapter 2 Determinant

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  1. Chapter 2Determinant

  2. Chapter 2 行列式 Determinant § 2-1 二階行列式 § 2-2 高階行列式 Define: A=nxn, A之determinant, det(A), 可遞迴定義如下： (1) If n=1, (2) If n=2, submatrix

  3. Property:

  4. If: By induction on n: “induction”歸納法” (2) 假設n=k成立，接下來要証明n=k+1也成立： consider n=k+1: Let

  5. § 2-3 行列式之性質 Note: (1) if A具有零列(行) (2) If A is upper triangular, then (3) If A is lower triangular, then (4) Property: 列運算對行列式的影響 example: hint: 作列運算至上三角,

  6. Property: Note: R:列基本矩陣； (4) If E: 基本矩陣 Lemma: A: nxn (1) if E: 列基本矩陣 (2) If E: 行…… if Lemma: E: 基本矩陣 if: Lemma: A: nxn, E:基本矩陣(不分行列) if: 純量可交換

  7. Theorem: if A: nxn, If: A: 可逆 A is nousingular Theorem: A: nxn, A:可逆 0 0 0 0 must be zero

  8. Theorem: (The most important theory is this section) If: (1) 若B可逆 (2)若B不可逆 (等號右邊永遠是0，所以只要 証明等號左邊一定是0就可) singular 只要証明AB是不可逆就可以了。 if AB不可逆，then AB is singular Note: are scalars, they can be interchanged. example:

  9. (5) if Note 8: where A is nxn example: (96成大資工) 証A is singular A: nxn, n: odd, 若 if: must be zero A is singular example: (96清大) if

  10. (Sol)

  11. example: (Vandermonde)很有名的矩陣 for example: if: By induction on n: (n至少要從n開始)

  12. Theorem: (“方塊矩陣的行列式”) 方塊上(下)三角的例子

  13. if: 如何拆求是issue！例如： Note:

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