Question

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

Answer 1

Given A² = A

So 7A – (I + A)³ 

= 7A – (I + 3A + 3A² + A³)

= 7A – I – 3A – 3 A² – A³

Given A² = A

7A – I – 3A – 3A – A³

= – I + A – A³

= – I + A – (A² × A)

= – I + A – (A × A)

= – I + A – A²

= – I + A – A = -I

So the value of 7A – (I + A)³ = – I.