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?
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.