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  4. If A 2= A and ( I + A )4 is equal to I + kA then find k.

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Q. If $A ^{2}= A$ and $( I + A )^{4}$ is equal to $I + kA$ then find $k$.

Matrices

Solution:

$(I+A)^{4}=(I+A)^{2} \cdot(I+A)^{2}$
$=\left(I+2 A+A^{2}\right) \cdot\left(I+2 A+A^{2}\right)$
$=(I+3 A) \cdot(I+3 A)$
$=I+6 A+9 A^{2}$
$=I+6 A+9 A$
$=I+15 A$