Matrices & Determinants
Matrices & Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
If $A$ is an idempotent matrix i.e. $A^2 = A$, and $B = I - A$, then which of the following is incorrect:
B^2 = B
B^2 = I
AB = 0
BA = 0
Step-by-Step Solution
Key Concept: Idempotent matrices ($A^2 = A$) satisfy the property that complementary projections $I - A$ also form orthogonal decompositions.
Given $A^2 = A$ and $B = I - A$, we compute $AB = A(I-A) = A - A^2 = A - A = 0$ and $BA = (I-A)A = A - A^2 = A - A = 0$. Also, $B^2 = (I-A)(I-A) = I - A - A + A^2 = I - 2A + A = I - A = B$. Thus both $A$ and $B$ are idempotent with $AB = BA = 0$.
Correct Answer: 2