Q. 445.0( 1 Vote )

If I is the ident

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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