If I is the identity matrix and A is a square matrix such A2 = A, then

1.	If I is the identity matrix and A is a square matrix such A2 = A, then what is the value of (I + A)2 – 3A?
Answer» We are given that, I is the identity matrix. A is a square matrix such that A² = A. We need to find the value of (I + A)² – 3A. We must understand what an identity matrix is. An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeroes. Take, (I+A)² – 3A = (I)²+ (A)² + 2(I)(A) – 3A [∵, by algebraic identity, (x+y)² = x² + y² + 2xy] ⇒(I+A)² – 3A = (I)(I) + A² + 2(IA) – 3A By property of matrix, (I)(I) = I IA = A ⇒(I+A)² – 3A = I + A² + 2A – 3A ⇒(I+A)² – 3A = I + A + 2A – 3A [∵, given in question, A² = A] ⇒ (I+A)² – 3A = I + 3A – 3A ⇒ (I+A)² – 3A = I + 0 ⇒ (I+A)² – 3A = I Thus, The value of (I+A)²– 3A = I.

If I is the identity matrix and A is a square matrix such A2 = A, then what is the value of (I + A)2 – 3A?

Answer»

We are given that,

I is the identity matrix.

A is a square matrix such that A² = A.

We need to find the value of (I + A)² – 3A.

We must understand what an identity matrix is.

An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeroes.

Take,

(I+A)² – 3A = (I)²+ (A)² + 2(I)(A) – 3A

[∵, by algebraic identity,

(x+y)² = x² + y² + 2xy]

⇒(I+A)² – 3A = (I)(I) + A² + 2(IA) – 3A

By property of matrix,

(I)(I) = I

IA = A

⇒(I+A)² – 3A = I + A² + 2A – 3A

⇒(I+A)² – 3A = I + A + 2A – 3A

[∵, given in question, A² = A]

⇒ (I+A)² – 3A = I + 3A – 3A

⇒ (I+A)² – 3A = I + 0

⇒ (I+A)² – 3A = I

Thus,

The value of (I+A)²– 3A = I.

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