Q. 24.7( 3 Votes )

# If a is a square

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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