1 markMCQSection A

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

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

The correct option is **D) 2B**. **Working:** - Given $AB = A$ and $BA = B$. - Compute $B^2$ using $B = BA$: $B^2 = (BA)B = B(AB) = B \cdot A = BA = B$. - Hence $B^2 = B$ (B is idempotent). - Therefore $B^2 + B = B + B = 2B$.

Algebra (Matrices)

Matrix Algebra — idempotent matrices

Options

(A)

2A

(B)

O

(C)

2I

(D)

2B

Official Answer

The correct option is D) 2B.

Working:

Given $AB = A$ and $BA = B$ .
Compute $B^2$ using $B = BA$ : $B^2 = (BA)B = B(AB) = B \cdot A = BA = B$ .
Hence $B^2 = B$ (B is idempotent).
Therefore $B^2 + B = B + B = 2B$ .

idempotent matrixAB = ABA = BassociativityB squared equals B2Bmatrix multiplication

Marking Scheme

11 mark: showing $B^2 = B$ and concluding $B^2 + B = 2B$ (option D).

Hint

Write $B = BA$ , then $B^2 = (BA)B$ and regroup as $B(AB)$ ; use $AB = A$ .

Quick Oral Answer

Substituting $B = BA$ gives $B^2 = B(AB) = BA = B$ , so B is idempotent and $B^2 + B = 2B$ .

Analysis & Explanation

This tests careful use of the associative property of matrix multiplication and the idempotent idea.

Concept:

A matrix M is idempotent if $M^2 = M$ . The two given relations force exactly this for B.
Key manipulation: substitute one factor using a given relation, then regroup with associativity.

Why D is correct:

$B^2 = (BA)B = B(AB) = BA = B$ , so $B^2 + B = 2B$ .

Why the distractors are wrong:

A ( $2A$ ): would need $B^2 = A$ , but $B^2 = B$ , not A.
B (O): would require $B^2 = -B$ , which the relations do not give.
C ( $2I$ ): assumes B is the identity; B is only idempotent, not necessarily I.

Common Mistakes

1Assuming $AB = A$ implies A is the identity or B is the identity — it does not.
2Treating matrix multiplication as commutative and writing $B^2 = AB = A$ .
3Stopping at $B^2 = B$ without adding B to reach $2B$ .

Interesting Facts

Idempotent matrices ( $M^2 = M$ ) are exactly the projection matrices used in statistics and computer graphics.

The pair $AB = A$ , $BA = B$ makes both A and B idempotent, and they act like mutual identities on each other's column space.

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Frequently Asked Questions

What does it mean that B is idempotent?

It means $B^2 = B$ — multiplying B by itself returns B. The conditions $AB=A$ and $BA=B$ force this property on B.

Can we conclude B is the identity matrix?

No. B is only idempotent. The identity is one special idempotent matrix, but many non-identity idempotent matrices exist, so $B^2=B$ does not make B equal to I.