Here we Don't need to calculate the Eigen values,
We just need the equation(cayley-Hamilton)
let K be the eigen value, then the relation AX=O we know
now
the matrix
1-k 0 3
2 1-k -1 =0
1 -1 1-k
=>(1-k){(1-k)2-1} -0(...) +3(2-(1-k))
=>simplifying we get
K3-3K2-K+9
now we don't need the value of K, according to cayley-hamilton, the matrix A ,matches this Equation..
so we can write
A3-3A2-A+9I=0 [I is identity matrix ]
so B is correct answer here