Q. 445.0( 1 Vote )

# If I is the ident

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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