Question

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.

Answer 1

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.A 2 – 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.