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I'm learning how to get the adjoint of a matrix using cofactors and the transpose, but how is this formula proven? When I try to google the proof for this, all I get is blog posts on how to get the adjoint of a matrix but not the proof of the formula itself.

Simon Suh
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  • I guess I'm just curious about the origin of the formula as in who discovered it and what its original purpose was. Like how did Einstain figure out E=mc2. I get that it's a definition, but still, there must have been some process to figuring it out and its relation to currently widely used matrix related formulas. – Simon Suh Jun 23 '22 at 20:52
  • don't want to be rude or anything, just really curious. :) – Simon Suh Jun 23 '22 at 21:01
  • The question is what motivates the definition — it’s a good question. – littleO Jun 23 '22 at 22:15
  • It’s funny that the “Don’t Memorise” channel you linked to presents the adjoint matrix in a way that is pure memorization, no motivation. – littleO Jun 23 '22 at 22:19

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Denoting the adjoint of a matrix $A$ by $A^\#$, it is well known that $$ AA^\# =\text{det}(A)\cdot I, $$ and in fact this is the main motivation for introducing the adjoint, since it leads to the formula for the inverse of $A$: $$ A^{-1}=\frac 1{\text{det}(A)}A^\#. $$ Now it also follows that $$ A^\#=\text{det}(A)A^{-1}.\tag{1} $$ This of course only works for invertible matrices but if you work out an appropriate way to compute the right-hand-side of (1) in terms of the entries of $A$, you will arrive at a formula for the adjoint which will miraculously be valid also for non-invertible matrices!

Ruy
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