If A is square matrix such

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 = I+ 3I2A + 3IA2 + A3

= I + 3IA + 3A+ AA2 (As I= I, n ∈ N)

= I + 3A + 3A + AA

= I + 3A + 3A + A2 = I + 3A + 3A + A = I + 7A

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