Why does the shortcut for finding inverse of 2x2 matrices work?

3,303

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 $A^{-1}$

Share:
3,303

Related videos on Youtube

Random person
Author by

Random person

Updated on July 23, 2022

Comments

  • Random person
    Random person over 1 year

    There is a short-cut for finding the inverse of 2x2 matrices

    Image of proof

    Can someone please explain why it works ? because I can't find any proof ?

    • Simply Beautiful Art
      Simply Beautiful Art almost 6 years
      Did you try multiplying the claimed result by $A$?
    • Random person
      Random person almost 6 years
      I know that it works, I want to know why it works
    • eranreches
      eranreches almost 6 years
      Well, it works because, it works. There are no hidden reasons here. It is just a simple calculation.
    • Simply Beautiful Art
      Simply Beautiful Art almost 6 years
      I'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.
    • Riley
      Riley almost 6 years
      The formula can be easily derived with Gauss Jordan
    • Simply Beautiful Art
      Simply Beautiful Art almost 6 years
      Oh man, mathmeeting.com, free homework help and solutions. Hm... just to let you know, our site policies are probably different than mathmeeting's.
    • Random person
      Random person almost 6 years
      I'm not asking for the answer to a homework question simply a reason why a short-cut I was taught works
    • Simply Beautiful Art
      Simply Beautiful Art almost 6 years
      Yes, 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.
  • Random person
    Random person almost 6 years
    Thanks gave me the best overview