Question 31 markMCQSection A

If AB=AAB = A and BA=BBA = B, then (B2+B)(B^2 + B) is equal to :

Algebra (Matrices)
Matrix Algebra — idempotent matrices

Options

(A)2A2A
(B)OO
(C)2I2I
(D)2B2B

Official answer

The correct option is D) 2B.


Working:


  • Given AB=AAB = A and BA=BBA = B.
  • Compute B2B^2 using B=BAB = BA: B2=(BA)B=B(AB)=B⋅A=BA=BB^2 = (BA)B = B(AB) = B \cdot A = BA = B.
  • Hence B2=BB^2 = B (B is idempotent).
  • Therefore B2+B=B+B=2BB^2 + B = B + B = 2B.
Answer keywords:idempotent matrixAB = ABA = BassociativityB squared equals B2Bmatrix multiplication

Marking scheme

  1. 1.1 mark: showing B2=BB^2 = B and concluding B2+B=2BB^2 + B = 2B (option D).

Hint

Write B=BAB = BA, then B2=(BA)BB^2 = (BA)B and regroup as B(AB)B(AB); use AB=AAB = A.

Quick oral answer

Substituting B=BAB = BA gives B2=B(AB)=BA=BB^2 = B(AB) = BA = B, so B is idempotent and B2+B=2BB^2 + B = 2B.

Common mistakes

  1. 1.Assuming AB=AAB = A implies A is the identity or B is the identity — it does not.
  2. 2.Treating matrix multiplication as commutative and writing B2=AB=AB^2 = AB = A.
  3. 3.Stopping at B2=BB^2 = B without adding B to reach 2B2B.

Analysis and explanation

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


Concept:


  • A matrix M is idempotent if M2=MM^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:


  • B2=(BA)B=B(AB)=BA=BB^2 = (BA)B = B(AB) = BA = B, so B2+B=2BB^2 + B = 2B.

Why the distractors are wrong:


  • A (2A2A): would need B2=AB^2 = A, but B2=BB^2 = B, not A.
  • B (O): would require B2=−BB^2 = -B, which the relations do not give.
  • C (2I2I): assumes B is the identity; B is only idempotent, not necessarily I.

Interesting facts

Idempotent matrices (M2=MM^2 = M) are exactly the projection matrices used in statistics and computer graphics.

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

Question details

Q3 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Algebra (Matrices)
Topic
Matrix Algebra — idempotent matrices
Bloom's level
analyze
Syllabus topic
Unit 2: Algebra — Matrices: multiplication and idempotent property
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What does it mean that B is idempotent?

It means B2=BB^2 = B — multiplying B by itself returns B. The conditions AB=AAB=A and BA=BBA=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 B2=BB^2=B does not make B equal to I.