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

Question

If A2=A, the (I+A)4 equals (A) I+15 A (B) I+7 A (C) I+8 A (D) I+11 A

Tardigrade Admin · Accepted Answer

I and A commute, we can apply binomial theorem to expand (I+A)4. We have (I+A)4 =I+4 A+6 A2+4 A3+A4 . text But A3 =A2=A, A4=A3=A ∴ (I+A)4 =I+4 A+6 A+4 A+A=I+15 A

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

$I + 15 A$

$I + 7 A$

$I + 8 A$

$I + 11 A$

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

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

I+15A

I+7A

I+8A

I+11A