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If $A$ and $B$ are two are square matrix such that $AB=I$ , then which of following are not necessarily not true

  1. $BA=I$
  2. $A=B^{-1}$
  3. $B=A^{-1}$
  4. $A^2=B$

$$AA^{-1}=I$$ $$BB^{-1}=I$$ I figured out that option $2,3$ are correct but how to check for $1$ and $4$

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    $\begingroup$ How could you verify 2. and 3. without proving 1. first? $\endgroup$
    – Claudius
    Aug 3, 2016 at 8:52
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    $\begingroup$ How does that answer my question? How did you verify 2. and 3.? $\endgroup$
    – Claudius
    Aug 3, 2016 at 8:55
  • $\begingroup$ @user218931 First line of my work show this $\endgroup$ Aug 3, 2016 at 9:16
  • $\begingroup$ $A^{-1}$ is by definition the unique matrix such that $AA^{-1} = I$ and $A^{-1}A = I$, i. e. both equalities have to be satisfied. This means that you first have to verify 1. in order to conclude 2. and 3. $\endgroup$
    – Claudius
    Aug 3, 2016 at 9:43
  • $\begingroup$ @user218931 Why it is necessary to first check 1 . I can put 2 and 3 in given equation and check whether it satisfy it not. $\endgroup$ Aug 3, 2016 at 14:13

1 Answer 1

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  1. note that A is full rank. for any x $$ABAx = Ax$$ $$A(BAx-x) = 0$$ that is $$BAx = x$$ $$ (BA-I)x = 0$$ $$BA = I$$
  2. left multiply both side by A $$A^3 = AB = I$$ which is not always true.
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