If A is square matrix such that A2 = A, then show that (I + A)3 =… - Vidyadip

Question

If A is square matrix such that A² = A, then show that (I + A)³ = 7A + I.

✓

Answer

We know that, A.I = I.A

So, A and I are commutative.

Therefore we can expand (I + A)³ like real number expansion.

So, (I + A)³ = I³ + 3I²A + 3IA² + A³

= I + 3IA + 3A² + AA² $($as Iⁿ = I, $\text{n}\in\text{N})$

= I + 3A + 3A + AA

= I + 3A + 3A + A²

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

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