Correct Answer - Option 4 :
\(\left[ \begin{matrix} 0 && 1 \\\ -2 && -3 \end{matrix} \right]\)
For a matrix A, whose eigen value is \(λ\) and corresponding eigen vector is X, C.E. equation is given by
AX=λ X
Assume, \(A= \begin{pmatrix} a & b\\ c & d \end{pmatrix}\)
For eigen value λ = -1⇒
\(\begin{pmatrix} a & b\\ c & d \end{pmatrix}\begin{pmatrix} +1 \\ -1 \end{pmatrix}=(-1)\begin{pmatrix} +1 \\ -1 \end{pmatrix} \)
\(a-b=-1\space\space\space\space\space\space \space...(i)\)
\(c-d=-1\space\space\space\space\space\space \space...(ii)\)
For eigen value λ = -2⇒
\(\begin{pmatrix} a & b\\ c & d \end{pmatrix}\begin{pmatrix} +1 \\ -2 \end{pmatrix}=(-2)\begin{pmatrix} +1 \\ -2 \end{pmatrix} \)
\(a-2b=-2\space\space\space\space\space\space \space...(ii)\)
\(c-2d=4\space\space\space\space\space\space \space...(iv) \)
From (i) - (iii)
b= -1+2 = 1
from (i)
a = 0
From (ii) - (iv)
d = -3
From (ii)
c = -2
So matrix \(A= \begin{pmatrix} 0 & 1\\ -2 & -3 \end{pmatrix} \)
