Question

If A is a square matrix such that $A^2 = I$, then $(A - I)^3 + (A + I)^3 - 7A$ is equal to

• A. $A$
• B. $I - A$
• C. $I + A$
• D. $3A$

Answer 1

Answer: (A)

$A$

Answer 2

$A^2 = I$ Now, $(A - I)^3 + (A + I)^3 - 7A$ $= A^3 - I^3 - 3A^2I + 3AI^2 + A^3 + I^3 + 3A^2I + 3AI^2 - 7A$ $= 2A^3 + 6AI^2 - 7A = 2A^2A + 6AI - 7A$ $= 2IA +6A - 7A = 2A + 6A - 7A = A \left[\because A^{2}=I\right]$