If a is a square matrix such that A2 = I, then find the simplified value of 
Given: a is a square matrix such that A2 = I
To find: the simplified value of (A – I)3 + (A + I)3 – 7A
Formula used:
(x + y)3 = x3 + y3 + 3xy(x + y)
(x – y)3 = x3 – y3 – 3xy(x – y)
(A – I)3 + (A + I)3 – 7A
= A3 – I3 – 3AI(A – I) + A3 + I3 + 3AI(A + I) – 7A
= A3 – I3 – 3A2I + 3AI2 + A3 + I3 + + 3A2I + 3AI2 – 7A
= 2A3 + 6AI2 – 7A
{∵ AI2 = A}
= 2A2.A + 6A – 7A
{∵ A2 = I}
= 2I.A – A
{∵ I.A = A}
= 2A – A
= A
Hence, simplified form of (A – I)3 + (A + I)3 – 7A = A
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