If AB = A and BA = B, then $(B^2 + B)$ is equal to :

Step 1: Analyze the given conditions

We are given two matrix equations: $AB = A$ and $BA = B$. These are the fundamental properties we will use to simplify the expression $B^2 + B$.

Step 2: Simplify $B^2$

We need to find the value of $B^2 + B$. Let's first simplify $B^2$. We can write $B^2$ as $B$ multiplied by $B$.

Step 3: Substitute using the given conditions

From the second given condition, we know that $B = BA$. We can substitute this into the expression for $B^2$.

Step 4: Apply associativity of matrix multiplication

Matrix multiplication is associative, so we can regroup the terms as $B(AB)$.

Step 5: Substitute again using the given conditions

From the first given condition, we know that $AB = A$. Substituting this into our expression gives $B^2 = BA$.

Step 6: Final substitution for $B^2$

Finally, from the second given condition, we know that $BA = B$. Therefore, we can conclude that $B^2 = B$.

Step 7: Calculate $B^2 + B$

Now that we have found $B^2 = B$, we can substitute this back into the original expression $B^2 + B$. This simplifies to $B + B$, which is $2B$.

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