Question

If A is a square matrix such that A² = I, then (A − I)³ + (A + I)³ − 7A is equal to 

पर्याय

• A

• I-A

• I+A

• 3A

Answer 1

\[\left( A - I \right)^3 + \left( A + I \right)^3 - 7A\]

\[ = A^3 - I^3 - 3 A^2 I + 3A I^2 + A^3 + I^3 + 3 A^2 I + 3A I^2 - 7A\]

\[ = 2 A^3 + 6A I^2 - 7A\]

\[ = 2A . A^2 + 6A - 7A\]

\[ = 2A . I - A \left( \because A^2 = I \right)\]

\[ = 2A - A\]

\[ = A\]

Hence, the correct option is (a).