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§2.2 行列式的应用

§2.2 行列式的应用. 行列式作为一中运算工具,在很多方面都有广泛的应用. 若把行初等变换施于 n 阶矩阵 A 上：. (1) 将 A 的某一行乘以数 k 得到 A 1 ，则 det A 1 = k (det A ) ； (2) 将 A 的某一行的 k (≠0) 倍加到另一行得到 A 2 , 则 det A 2 = det A ； (3) 交换 A 的两行得到 A 3 , 则 det A 3 = - det A. 证. (1) 按乘以数 k 的那一行展开，即得结论成立。. (2). 例 1 奇数阶反对称阵的行列式必为零.

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§2.2 行列式的应用

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  1. §2.2行列式的应用 行列式作为一中运算工具,在很多方面都有广泛的应用

  2. 若把行初等变换施于n阶矩阵A上： (1) 将A的某一行乘以数k得到A1，则 detA1 = k(detA)； (2) 将A的某一行的k(≠0)倍加到另一行得到A2 ,则 detA2 = detA； (3) 交换A的两行得到A3, 则detA3 = - detA. 证 (1)按乘以数k的那一行展开，即得结论成立。

  3. (2)

  4. 例1奇数阶反对称阵的行列式必为零. 证Ann (n为奇数)满足：

  5. 1 伴随矩阵 定义6 (伴随矩阵)

  6. 定理1 证：

  7. 引理 1 设L有如下分块形式的(n+m)阶矩阵: 证明：略 引理 2 设A、B皆为n阶矩阵，则有 证明：略

  8. 定理2(方阵可逆的充要条件) 证：

  9. 例 2 判定下列方阵是否可逆，若可逆，求其逆矩阵 解

  10. 推论 证：

  11. 例3 解

  12. 例4 证：

  13. 例 5 证

  14. 例6设 解 = n！

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