+1 vote
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Let $A$ be a $3\times 3$ matrix with Eigen values $-1,1,0$.Then $\mid A^{100}+I\mid$ is...

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+3

Eigen value of A (-1,1,0) given. So, the eigen value of A^100 will be [ (-1)^100 , (1)^100, (0)^100 which is (1,1,0) ] hence the Eigen value of matrix --> [ A^100+ I ] = ( 1+1 , 1+1 ,0+1 ) = (2,2,1) . As it is asking for  |A^100+I| = Product of it's Eigen value = ( 2*2*1 ) = 4 .

+1 vote
EIGEN VALUES OF A^(100) will be (-1)^(100)=1

(1)^(100) =1 , (0)^(100)=0, det|A^(100)+ I| = 2*2*1 =4
by Loyal