线性代数中行列式展开定理的教学难点及解决方案

Difficulties and Effective Solutions in Teaching Determinant Expansion Theorem in Linear Algebra

行列式的展开定理在线性代数中具有非常重要的基础性地位,克莱姆法则、可逆矩阵的判定和矩阵的秩都依赖于这一定理.不同于行列式的定义和基本性质,行列式按行展开的定理对学生们而言比较困难,尤其是其证明过程,如果学生不懂其证明思路,就很难真正理解定理的意义,并直接影响问题的解决.本文将从教材、教学、练习这3个重要环节分析学生的实际困难在哪里,并给出解决教学难点的有效的解决办法,并且本文所提出的方法也完全适用于其他类似课程的教学.

The expansion theorem of the determinant has a very important foundational position in Linear Algebra,for example,Cramer's rule,the judgment of invertible matrices,and the rank of the matrix all depend on this theorem.Different from the definition and basic properties of the determinant,the theorem that the determinant expands by rows is more difficult for students,especially the proof process.If students do not understand the proof idea,it is difficult to understand the meaning of the theorem truly and affect problem solving directly.The actual difficulties of students from the three important links of teaching materials,teaching,and exercises will be analyzed,and an effective solution be given to solve the teaching difficulties,and the methods proposed in this article are also fully applicable for teaching other similar courses.