Question

(If A is a square matrix such that A² = I,then find the value of (A − I)³ + (A+I)³ −7A .

Answer 1

(a) We have, A2 = I

Also A and I are commutative, so we can expand (A+I)n using expansion of (a+b)n, where a and b ∈ C

∴ (A-I)3 + (A+I)3 - 7A

= A3 - 3A2 + 3A - I3 + A3 + 3A2 + 3A + I3 - 7A

= 2A3 + 6A - 7A

= 2A2 . A + 6A - 7A

= 2I.A + 6A - 7A

= 2A + 6A - 7A = 8A - 7A = A