Why does the shortcut for finding inverse of 2x2 matrices work?
Solution 1
The r.h.s. is the adjugate matrix of $A$, i.e. the transpose of the matrix of cofactors of $A$, which is easy to calculate in the case of $2\times2$ matrices.
One proves, independently of the size of the matrix, that $$A\cdot\operatorname{adj}A=(\det A)\,I.$$ This formula is a rewriting of Laplace's formula, and is valid for matrices over any commutative ring.
Solution 2
If $A$ is an invertible $2 \times 2$ matrix, then it satisfies:
$$A A^{-1} = I$$
Let $A = \begin{bmatrix} a & b\\ c & d \end{bmatrix}$. Then you need to solve the augmented system:
$$ \left[ \begin{array}{cc|cc} a&b&1&0\\ c&d&0&1\\ \end{array} \right] $$
Now row-reduce this system until the left side becomes $I$. The right side will then give you the desired $A^{-1}$
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Updated on July 23, 2022Comments
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Random person over 1 year
There is a short-cut for finding the inverse of 2x2 matrices
Can someone please explain why it works ? because I can't find any proof ?
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Simply Beautiful Art almost 6 yearsDid you try multiplying the claimed result by $A$?
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Random person almost 6 yearsI know that it works, I want to know why it works
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eranreches almost 6 yearsWell, it works because, it works. There are no hidden reasons here. It is just a simple calculation.
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Simply Beautiful Art almost 6 yearsI'm not sure I understand your question. "why it works" isn't a very good question, especially if you swat away suggestions such as my first comment.
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Riley almost 6 yearsThe formula can be easily derived with Gauss Jordan
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Simply Beautiful Art almost 6 yearsOh man, mathmeeting.com, free homework help and solutions. Hm... just to let you know, our site policies are probably different than mathmeeting's.
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Random person almost 6 yearsI'm not asking for the answer to a homework question simply a reason why a short-cut I was taught works
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Simply Beautiful Art almost 6 yearsYes, I know. But since this seems to be your first question, I wanted to point you towards our help center. Note that your question may be downvoted and closed due to a lack of effort/context; this site tends to encourage questions that show thought has been put into them.
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Random person almost 6 yearsThanks gave me the best overview