If A is a square matrix such that A2 = A, then write the value of 7A–(I + A)3, where I is the identity matrix.
We are given that,
A is a square matrix such that,
A2 = A
I is an identity matrix.
We need to find the value of 7A – (I + A)3.
Take,
7A – (I + A)3 = 7A – (I3 + A3 + 3I2A + 3IA2)
[∵, by algebraic identity, (x + y)3 = x3 + y3 + 3x2y + 3xy2]
⇒ 7A – (I + A)3 = 7A – I3 – A3 – 3I2A – 3IA2
⇒ 7A – (I + A)3 = 7A – I – A3 – 3I2A – 3IA2
⇒ 7A – (I + A)3 = 7A – I – A.A2 – 3I2A – 3IA2
⇒ 7A – (I + A)3 = 7A – I – A.A2 – 3A – 3A2
[∵, by property of identity matrix,
I2A = A & IA2 = A2]
⇒ 7A – (I + A)3 = 7A – I – A.A – 3A – 3A
[∵, it is given that, A2 = A]
⇒ 7A – (I + A)3 = 7A – I – A2 – 6A
[∵, A.A = A2]
⇒ 7A – (I + A)3 = 7A – I – A – 6A
[∵, it is given that, A2 = A]
⇒ 7A – (I + A)3 = 7A – I – 7A
⇒ 7A – (I + A)3 = -I
Thus, the value of 7A – (I + A)3 is –I.