If A is given matrix and 'B' is correspanding eigen vector then multiplication of AB=(k)B.then 'k' is correspanding eigen value.
so check this condition for all options wii given eigen value of the eigen vector.
check option 1:
A=$\begin{bmatrix} 1 &-1 &1 \\ 1& -1&1 \\ -1 &1 &1 \end{bmatrix}$.
B=$\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$.
AB=$\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$.// so eigen value is =1.
check option 2:
AB=$\begin{bmatrix} 0\\ 0\\ 2 \end{bmatrix}$ // no eigen value.
check option 3:
AB=$\begin{bmatrix} 2\\ 2\\ 0 \end{bmatrix}$ // no eigen value.
check option 4:
AB=$\begin{bmatrix} -1/2\\ -1/2\\ 1/2 \end{bmatrix}$ //NO EIGEN VALUE EXIST.
finally only on eigen value correspanding eigen vector. i.e)eigen value=1.
