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Let $A$ be a square matrix. Another square matrix $B$ is called inverse of $A$ if $$AB=BA=I.$$ My question is whether just $AB=I$ or $BA=I$ is not sufficient to call $B$ as the inverse of $A$? If it is not, then give a counter example where $AB=I$, but $AB\ne BA.$

José Carlos Santos
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Ranveer Singh
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    See https://math.stackexchange.com/questions/2337572/are-there-matrices-such-that-ab-i-and-ba-neq-i/2337575#2337575 – Dave Jul 30 '17 at 14:31

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$AB=I\implies \det A\neq 0\implies A$ is invertible.

$\implies B = A^{-1}$

$\implies BA = I$

Shuri2060
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Since invertible square matrices over a field $K$ form a group, namely $GL_n(K)$, the left-inverse is equal to the right-inverse - see this duplicate.

Dietrich Burde
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