If AB = A and BA = B, then (B2+B) is equal to :
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Step-by-Step Solution
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 B2+B.
Step 2: Simplify B2
We need to find the value of B2+B. Let's first simplify B2. We can write B2 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 B2.
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 B2=BA.
Step 6: Final substitution for B2
Finally, from the second given condition, we know that BA=B. Therefore, we can conclude that B2=B.
Step 7: Calculate B2+B
Now that we have found B2=B, we can substitute this back into the original expression B2+B. This simplifies to B + B, which is 2B.