The Application Of The Theory Of Equivalent Systems Of Linear Equations In The Proof Of Cramer's Rule

There is an oversight in the reference book (the first semester of high school sophomore, page 112) for mathematics textbook of Shanghai edition:for a system of linear equations involving three variables denote determinant if D= Dx=Dy=Dz=0, then (Ⅰ) has infinitely many solutions.But actually, when D= Dx-Dy= Dz= 0, (Ⅰ) does not only can have infinitely many solutions, but also can have no solution. The article points out this oversight, and analyzes the intrinsic reasons as follows:(Ⅰ) was transformed into in the page 100 and 101 of the textbook, however, the transformation may not guarantee (I*) has the same solution as the original one. Unfortunately, the authors of the reference book might not realize the fact, as a result, they thought that since (I*) had infinitely many solutions when D=Dx=Dy=Dz=0, so did (Ⅰ) under the same condition.Furthermore, the article analyzes the reason why the transformation may not guarantee (Ⅰ) and (I*) have the same solution, points out that (Ⅰ)'s solution set is contained in (Ⅰ*)'s and obtains a conclusion:(1) if D≠0,then (I*) and (I) both have unique solution;(2) if D=0 and at least one of Dx, Dy and Dz is not zero, then (I*) and (Ⅰ) have no solution;(3) if D= Dx=Dy=Dz=0, then (Ⅰ) can have infinitely many solutions and can also have no solution, while (I*) has infinitely many solutions.At the end of the article, several improved schemes for the oversight are provided.