Question:
If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I.
Solution:
We know that,
A . I = I . A
So, A and I are commutative.
Thus, we can expand (I + A)3 like real numbers expansion.
So, (I + A)3 = I3 + 3I2A + 3IA2 + A3
= I + 3IA + 3A2 + AA2 (As In = I, n ∈ N)
= I + 3A + 3A + AA
= I + 3A + 3A + A2 = I + 3A + 3A + A = I + 7A