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Given is the matrix $A=\begin{pmatrix} a & b\\ c & d \end{pmatrix}$. Derive a (necessary and sufficient) condition (equation or inequation in the coefficients $a,b,c,d$) such that $A$ is regular and determine $A^{-1}$.

I'm not quite sure how such a condition is supposed to look like with these mentioned coefficients.

So I know that $\det(A) = 0$ is a necessary and sufficient condition for a matrix to be not invertible e.g. not regular. (Wikipedia)

From this it follows that $\det(A) \neq 0$ is a necessary and sufficient condition for a matrix to be invertible e.g. regular.

How do we achieve that?

  1. If $a=0$ and/or $d=0$, then $c,b \neq 0$
  2. If $c=0$ and/or $b=0$, then $a,d \neq 0$
  3. $a \cdot d \neq c \cdot b$

So this would be a necessary and sufficient condition? Not really, in my opinion.. :(

The two sentences at the beginning seemed better but I didn't use any coefficients there as mentioned in the example task.

How do you do this correctly? I'm really scared of such a task in a test..

eyesima
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    Assuming that $ad - bc \ne 0$, you can explicitly write down an inverse of $A$. If $ad - bc = 0$, then you can explicitly write down a vector $x \ne 0$ such that $Ax = 0$. –  Apr 18 '18 at 19:14
  • @user296602 Then the first you mentioned $ad-bc \neq 0$ should be the sufficient condition because this must be the case such that an inverse exists, right? But then what's the neccessary condition? – eyesima Apr 18 '18 at 19:30
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    It is necessary and sufficient if you follow the two parts my comment. –  Apr 18 '18 at 19:32

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