# Thread: AB = I_n can we obtain BA =I_n

1. ## AB = I_n can we obtain BA =I_n

Let A,B are square matrices n x n; and I_n is identity matrix.
If AB =I_n, can we obtain BA =I_n?

2. Originally Posted by mahefo
Let A,B are square matrices n x n; and I_n is identity matrix.
If AB =I_n, can we obtain BA =I_n?
Yes, since A and B are inverses of each other.

3. I know the definition of inverse matrix.
Can you show me the way from AB=I_n to get BA=I_n?
Thanks a lot.

4. Originally Posted by mahefo
I know the definition of inverse matrix.
Can you show me the way from AB=I_n to get BA=I_n?
Thanks a lot.

Ok, then you hopefully know that the product of two matrices is invertible iff every one of them is invertible, so in our case both matrices are invertible, but then:

$\displaystyle BA=A^{-1}(AB)A=A^{-1}\cdot I\cdot A = A^{-1}A=I$

Tonio

5. Originally Posted by tonio

$\displaystyle BA=A^{-1}(AB)A$
well, that is not quite right because what we have is that $\displaystyle A$ has a right inverse. we don't know yet if the left inverse of $\displaystyle A$ exists or not.

Originally Posted by mahefo
Let A,B are square matrices n x n; and I_n is identity matrix.
If AB =I_n, can we obtain BA =I_n?