Question

If A is a square matrix such that A2=I then A‚àí1 is equal to

Answer 1

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