To determine if the product $AB$ of two idempotent matrices $A$ and $B$ is idempotent, let us explore the properties of idempotent matrices.

Definition of an Idempotent Matrix

A matrix $A$ is idempotent if: $A^2 = A$ Similarly, matrix $B$ is idempotent if: $B^2 = B$

Condition for $AB$ to Be Idempotent

For $AB$ to be idempotent, it must satisfy: $(AB)^2 = AB$

Expanding $(AB)^2$: $(AB)^2 = AB \cdot AB = A (BA) B$

Thus, $AB$ is idempotent if: $A (BA) B = AB$

Analysis

For the equality $A (BA) B = AB$ to hold, specific conditions must exist on $A$ and $B$. In general, this is not true for arbitrary idempotent matrices $A$ and $B$, because $BA \neq AB$ in general, and the intermediate term $BA$ does not simplify unless $A$ and $B$ satisfy additional constraints.

Conclusion

The product $AB$ of two idempotent matrices $A$ and $B$ is not guaranteed to be idempotent in general. However, there are special cases where it could be idempotent, such as when $A$ and $B$ commute ($AB = BA$) or have specific structural relationships.

Would you like a deeper exploration of examples or conditions for specific cases?

Related Questions:

1. What are examples of idempotent matrices?
2. Can you prove that $AB$ is idempotent if $A$ and $B$ commute?
3. How does idempotency relate to eigenvalues and eigenvectors of matrices?
4. What are applications of idempotent matrices in linear algebra?
5. How does the idempotency of a matrix relate to its rank?

Tip:

To test if $AB$ is idempotent in a specific case, compute $(AB)^2$ and compare it to $AB$ directly.