Chapter 4 is the most theoretical chapter in the book in the sense that it is entirely dedicated to proving that a certain computational result cannot be achieved--namely, a diagonal square matrix cannot be transformed by additions or subtractions of adjacent rows or columns into another diagonal matrix in which the product of the diagonal entries is changed. The explorations of Chapter 2 already made the likelihood of there being such a transformation of diagonal matrices seem small, but no amount of trial-and-error can prove an impossibility statement of this kind. One could always choose to believe that a transformation of the required type existed but that one had not yet been clever enough or been able to treat a complicated enough example to find one. As in any proof that something is impossible--for example, the proof that an angle cannot be trisected by a ruler-and-compass construction--an abstract argument is necessary.
The argument is the one given in Chapter 4: There is a formula assigning numbers to square matrices in such a way that (1) equivalent matrices are assigned the same number and (2) diagonal matrices are assigned the product of their diagonal entries.
Matlab can quickly find the determinant of a square matrix--all you need to do is enter the matrix A and then enter det(A) at the Matlab prompt--but this is of no help in proving the theorem that a matrix has only one determinant, because the fact that the number det(A) returned by Matlab has properties (1) and (2) cannot be proved by computing values. What is needed is an analysis of the way the determinant is found, and Matlab simply gives the result without any explanation.
In short, there seems to be no worthwhile way to use Matlab to explain the content of this chapter. If you are unwilling--and you should be--to accept on faith that a matrix has only one determinant, you must study the proof.
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