高等代数在实际生活中的应用

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摘要：随着现代科学的发展，数学在经济中广泛而深入的应用是当前经济学最为深刻的因素之一，马克思曾说过：“一门学科只有成功地应用了数学时，才真正达到了完善的地步”。下面通过具体的例子来说明高等代数知识在经济生活中的应用。关键词：矩阵；特征值；经济应用一：在经济生活中的应用1．“活用”行列式定义定义：用符号表示的n阶行列式D指的是n!项代数和，这些项是一切可能的取自D不同行与不同列上的n个元素的乘积的符号为。由定义可以看出。n阶行列式是由n!项组成的，且每一项为来自于D中不同行不同列的n个元素乘积。实例1：某市打算在第“十一”五年规划对三座污水处理厂进行技术改造，以达到国家标准要求。该市让中标的三个公司对每座污水处理厂技术改造费用进行报价承包，见下列表格(以1万元人民币为单位)．在这期间每个公司只能对一座污水处理厂进行技术改造，因此该市必须把三座污水处理厂指派给不同公司，为了使报价的总和最小，应指定哪个公司承包哪一座污水处理厂?用．在中学数学中．高次多项式的因式分解比较困难．可以将多项式的因式分解转化为矩阵的运算．利用矩阵的乘法、矩阵的秩等相关的知识来分解整系数多项式，并将计算过程设计成相应的pascal程序语言．n次整系数多项式的因式分解在有理数域中可以利用艾森斯坦因(Eisenstein)判别法，先判别n次整系数多项式是否可约，如果可约，就先求可能的有理根。再初步过滤，然后用综合除法确认．下面给出利用矩阵分解整系数多项式的方法．分解定理我们已经在高等代数教材的前几章学过，以下着重介绍程序的应用。定理是n次整系数多项式，则次整因式的充分必要条件是存在一个系数为整数且秩为1的(r+1)*(n-r+1)阶