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consider $x = \begin{bmatrix}a & b\\c & d\end{bmatrix}$

when finding the inverse of the matrix why do we specifically flip the diagonals, and in particular make $b$ and $c$ negative? what does this represent geometrically?

user1729
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Arsh Dixit
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    We do it because it works. – José Carlos Santos Aug 02 '21 at 18:37
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    This is of no importance - geometrical or otherwise. You only do this for 2x2 matrices, because that's what the algorithm for finding the inverse produces for 2x2 matrices. – Stefan Octavian Aug 02 '21 at 18:40
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    See this MO-post, for a geometric explanation in terms of Iwasawa decomposition. – Dietrich Burde Aug 02 '21 at 18:41
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    The more general $n\times n$ algorithm that $A^{-1} = \frac1{\det A}\left[\begin{smallmatrix}d&-c\-b&a\end{smallmatrix}\right]$ is a manifestation of goes via the adjugate matrix. – Arthur Aug 02 '21 at 18:44
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    @Arthur Yes, this is exactly Qiaochu Yuan's answer at the MO-post. – Dietrich Burde Aug 02 '21 at 18:46
  • @DietrichBurde Right, I was too busy formatting my smallmatrix to see your link. – Arthur Aug 02 '21 at 18:47
  • @Arthur No problem, I just wanted to support your comment here. – Dietrich Burde Aug 02 '21 at 18:48
  • In terms of the geometry of the matrix (viewed as a 2-dimensional array of numbers), the transpose of the matrix of cofactors is what you get by swapping the diagonal entries and negating the antidiagonal entries. This is useless geometrically and not very useful mnemonically, except, if (as I did) you found that what goes on in the $2 \times 2$ case is so bizarre as to fix itself in your memory, and that helps you remember the rule for inverting larger matrices using the transpose of the matrix of cofactors. – Rob Arthan Aug 02 '21 at 20:36
  • One nice explanation is given in this video from 3blue1brown about Cramer's rule. – Ben Grossmann Aug 02 '21 at 22:46
  • Hey, I really appreciate you guys sending me these resources but I should have mentioned that I just got out of high school and can't seem to understand any of them, pretty much just learning about what a matrix is. – Arsh Dixit Aug 03 '21 at 07:31
  • In this case your question "what does this represent geometrically?" is perhaps too early, and you should first follow matrices and linear algebra of its own. The very first comment seems to be the best answer then, or not? – Dietrich Burde Aug 03 '21 at 08:55
  • yeah in a sense, I don't regret asking though, I will complete linear algebra and see, maybe mess around with it by myself – Arsh Dixit Aug 03 '21 at 11:01

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