Why $BA +AB = A+B$?

Question

Given $BA = A$ and $AB = B$ and $A$ and $B$ are two square matrices, why does $BA +AB = A+B$ ?

Please explain.

ADDED:

The actual problem is it was given that,$BA = A$ and $AB = B$ hence find the value of $A^2 + B^2$.

I simply utilized the fact that if $BA = A$ and $AB = B$ then $A$ and $B$ are idempotent matrices, hence $A^2 + B^2 = A + B$ but the module solution is something more algebraic they break $A^2 + B^2$ and then performs usual substitution after that they showed the result but I don't understand how $BA +AB = A+B$.

Erm. Isnt this just a trivial matter of substituting in the given conditions? — George Lowther, Jan 15 '11 at 15:13
As you edited, $BA=A$ and $AB=B$, is a given of the problem, not something to be derived. But $BA=A$ doesn't make either one idempotent. — Ross Millikan, Jan 15 '11 at 15:27

score 3 · Answer 1 · answered Jan 15 '11 at 14:55

3

This isn't true in general. For example take $A$ as the zero matrix and $B$ as the identity matrix. Then the left hand side is just the zero matrix and the right hand side is the identity matrix, which are not the same.

answered Jan 15 '11 at 14:55

Adrián Barquero

15,124

Fixed. – Quixotic Jan 15 '11 at 14:59

score 3 · Answer 2 · answered Jan 15 '11 at 14:55

3

This is not true, indeed consider the case when $A = 0 $ is the zero matrix and $B= Id$ is the identity, then LHS is the zero matrix but RHS is the identity.

answered Jan 15 '11 at 14:55

milcak

4,129

Why $BA +AB = A+B$?

2 Answers2