Talk:Matrix Product with Adjugate Matrix

I'm sorry to say it, but all these pages about determinants and invertibility of matrices are terribly ill-formulated:


 * Inverse of Matrix gives the inverse assuming it exists, but in Matrix is Invertible iff Determinant has Multiplicative Inverse it is claimed that it shows that invertible determinant implies invertible matrix. (And this is the main theorem on which pretty much everyting in Linear Algebra depends!). Don't understand the problem? Look again.
 * Much better way to proceed: write a page Matrix Product With Adjugate Matrix which shows the general (very frequently used!!!!) $A\cdot \operatorname{adj}(A)=\det(A)\cdot I$. From here, theorems on invertibility can be properly stated.
 * Problems: For some reason (see conclusion below) all of the concepts used in the derivation of the crucial identity $A\cdot \operatorname{adj}(A)=\det(A)\cdot I$ are defined for determinants instead of matrices: Definition:Cofactor, Definition:Adjugate. Horrific.
 * A possible explanation could be that some references are based on the old-fashioned theory of determinants, which existed before matrices. But if ProofWiki decides to hold to that philosophy, it is doomed (and I'm not exaggerating).

Conclusion: Don't blindly trust books! (Especially not books that are merely on computer science or physics than on mathematics.) Authors too make mistakes, or weird choices. Mathematics has developed a lot since some books were written. Assuming they're right or including the entirety of existing conventions in ProofWiki makes it VERY HARD to build on it. --barto (talk) 16:14, 3 February 2017 (EST)


 * Permission to be bold and do something about it, please. :| --barto (talk) 16:18, 3 February 2017 (EST)


 * As long as the something-about-it sets up an equivalent set of definitions / theorems and proves equivalence -- or in some way connects the existing stuff in. The existing material is worth keeping, if only to ensure that people familiar with the old bad ways have a route into the new good ways.


 * If the old stuff is seriously wrong, then keep the pages with a succinct description (structured rigorously like a proof if appropriate) explaining why. If it's just that the thinking is loose and non-rigorous, then fill the gaps. Whatever the problem is, my approach would be to raise a page about it and explain.


 * But this is outside my limits and I don't know the details. Whatever you know about this stuff which supersedes what's here (and can back it up by literature), feel free to take it on. --prime mover (talk) 17:24, 3 February 2017 (EST)