Matrices & Determinants
Non-singular idempotent matrix properties
MMTS_Full_Test_15
Grade 12
Question:
Let $A$ be a non-singular idempotent matrix of order $2025\times2025$. Consider statements: (i) Trace of $A$ = 2025, (ii) $A$ has to be a scalar matrix, (iii) Trace of adjoint of $A^2$ = 2025. Which are true?
(A) (i) and (ii)
(B) (ii) and (iii)
(C) (i), (ii) and (iii)
(D) none of these
Step-by-Step Solution
Key Concept: $A^2=A$ and $A$ non-singular $\Rightarrow A=I$. Trace$(I)=2025$ ✓. $I$ is scalar ✓. $A^2=I$, adj$(I)=I$, trace $=2025$ ✓.
$A=I$, all three statements hold.
Correct Answer: (C) (i), (ii) and (iii)