If A is square matrix such that A^2 = A, then show that (I + A)^3… - Vidyadip

Question

If $A$ is square matrix such that $A^2 = A,$ 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, $(I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3$^
$= I + 3IA + 3A^2 + AA^2 ($as $I^n = I, \text{n}\in\text{N})$
$= I + 3A + 3A + AA$
$= I + 3A + 3A + A^2$
$= I + 3A + 3A + A = I + 7A$

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