Matrices and Determinants
Algebra of Matrices
IIT-JEE 2004
Grade 12
Question:
If $A$ is a square matrix such that $A^2 = A$, then $(I + A)^3 - 7A$ equals:
(1) $I$
(2) $A$
(3) $2A$
(4) $O$
Step-by-Step Solution
Key Concept: Expand $(I + A)^3 = I + 3A + 3A^2 + A^3$ using the binomial expansion (valid since $I$ commutes with $A$), and then repeatedly replace $A^2$ by $A$ using the given condition $A^2 = A$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)