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  4. If A2=A, the (I+A)4 equals

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Q. If $A^2=A$, the $(I+A)^4$ equals

Matrices

Solution:

$I$ and $A$ commute, we can apply binomial theorem to expand $(I+A)^4$. We have
$ (I+A)^4 =I+4 A+6 A^2+4 A^3+A^4 . $
$\text { But } A^3 =A^2=A, A^4=A^3=A $
$\therefore (I+A)^4 =I+4 A+6 A+4 A+A=I+15 A$