If matrix $AB=A$, does it mean B must be an identity matrix?

Question

If matrix multiplication $AB=A$, does it mean $B$ must be an identity matrix? If not, why? What conditions?

$A$ is not a zero matrix.

Answer 1

$AB=A$ means $A(B-I)=0$ so the condition is precisely that the image of $B-I$ is contained in $\ker A$. Unless $A$ is injective, this does not force $B=I$.

Answer 2

Another example is to consider the case : A= B

=> $A^2$ = A , so if A is a projector (which is not the identity), you can find that B=A verify your equality, while B is not the identity

In fact, if you let C be the projection on Ker(A), and B= I+C

AB=A(I+C) = A +A*C =A . If A is not iversible, C is not 0, and B is not the identity.

Answer 3

Let $$ A=B=\left(\begin{matrix} 1&0\\ 0&0\end{matrix}\right) $$ then $$ AB=\left(\begin{matrix} 1&0\\ 0&0\end{matrix}\right). $$

Nevertheless, if $A$ is non-singular, then $B=I$. If $A$ is singular, then we can always find an example such us the one above.

Answer 4

The equality $AB = A$ holds if and only if each row of $A$ is either an all zero row or an eigenvector with eigenvalue $1$ for $B.$ Hence, as long as $B$ has $1$ as an eigenvalue, we can construct a matrix $A$ of rank $1$ with $AB = A.$