Q3 of 45 Page 1

If A is a square matrix such that A2 = A, then write the value of 7A - (I + A)3, where I is an identity matrix.

Given: A2 = A


To find 7A - (I + A)3


Using (a + b)3 = a3 + b3 + 3a2b + 3ab2


⇒ 7A – (I + A)3 = 7A – (I3 + A3 + 3(I)2A + 3(I)A2)


⇒ 7A – (I + A)3 = 7A – (I3 + A2A + 3I2A + 3IA2)


As I is identity matrix I3 = I and given A2 = A


⇒ 7A – (I + A)3 = 7A – (I + AA + 3A + 3A)


⇒ 7A – (I + A)3 = 7A – (I + A2 + 6A)


⇒ 7A – (I + A)3 = 7A – (I + A + 6A)


⇒ 7A – (I + A)3 = 7A – (I + 7A)


⇒ 7A – (I + A)3 = 7A – I – 7A


⇒ 7A – (I + A)3 = –I


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