5
$\begingroup$

If B and C are both inverses of the matrix A,then B=C.

Can't i prove it in following way ?

Proof:

AB=BA=I and AC=CA=I,then BA=CA=I

By postmultiplication $\Rightarrow (BA)(A^{-1})=(CA)(A^{-1})=(I)(A^{-1})\Rightarrow B=C=A^{-1}$,

or by premultiplication $AB=AC=I\Rightarrow (A^{-1})(AB)=(A^{-1})(AC)=(A^{-1})(I)\Rightarrow B=C=A^{-1}$.

$\endgroup$
3
  • 5
    $\begingroup$ What does $A^{-1}$ mean before uniqueness of the inverse has been established? $\endgroup$ Feb 3, 2015 at 12:33
  • $\begingroup$ @AlgebraicPavel Actually the proof is available in any linear algebra book. My main intention was to prove it by myself. As it was not the same as in the book, i wanted to check whether it makes sense. $\endgroup$
    – time
    Feb 3, 2015 at 13:04
  • 1
    $\begingroup$ Yes but it's unnecessarily complicated since your proof assumes the existence of three inverses while two are enough. $\endgroup$ Feb 3, 2015 at 13:09

1 Answer 1

11
$\begingroup$

There is much much simpler.

$B=BI=B(AC)=(BA)C=IC=C$

$\endgroup$
0

Not the answer you're looking for? Browse other questions tagged .