For square matrices $A$, $B$, is $AB=I$ sufficient that $A$ and $B$ are inverse of each other?

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If $AB = I$ then $BA = I$

If $A$ and $B$ are two square matrices, and we know $AB=I$ where $I$ is the identity matrix. Is it sufficient that $BA=I$ as well so that $A$ and $B$ are inverse matrices of each other?

Just found out that this is a duplicate question.

@EuYu Good to know that. Thank you. – JACKY88 Dec 12 '12 at 04:53 — JACKY88, Dec 12 '12 at 04:53

score 4 · Accepted Answer · answered Dec 12 '12 at 04:40

4

In fact, something stronger is true. Suppose that $A$ is invertible, and that $AB=I$. Then, $B=A^{-1}$ so that $BA=A^{-1}A=I$. Now, if $AB=I$, then $\det(A)\det(B)=1$ so that $\det(A)\ne 0$. So, $A$ is invertible. From this, you see that if just $AB=I$ then $A$ is invertible and $B=A^{-1}$.

answered Dec 12 '12 at 04:40

Alex Youcis

54,059

Just to comment that this only works for square matrices of finite order. – Ittay Weiss Dec 12 '12 at 07:07

For square matrices $A$, $B$, is $AB=I$ sufficient that $A$ and $B$ are inverse of each other?

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