Question 361049
For λ = -1, replacing -1 into the homogeneous system A - λI = O, we get the augmented homogeneous system
{{{(matrix(3,4, 2,-1,2,0,0,3,0,0,3,-3,3,0))}}}.
~{{{(matrix(3,4, 2,-1,2,0,0,1,0,0,1,-1,1,0))}}}Divide r2 and r3 by 3.
~{{{(matrix(3,4, 2,-1,2,0,0,1,0,0,1,0,1,0))}}}Add row2 to r3
~{{{(matrix(3,4, 0,-1,0,0,0,1,0,0,1,0,1,0))}}}-2*r3 + r1
~{{{(matrix(3,4, 0,0,0,0,0,1,0,0,1,0,1,0))}}}Add r2 to r1
~{{{(matrix(3,4, 1,0,1,0,0,1,0,0,0,0,0,0))}}}Interchange r1 and r3.
Thus y = 0 and x + z = 0, or z = -x. hence,
{{{(matrix(3,1,x,y,z))=(matrix(3,1,x,0,-x)) = x(matrix(3,1,1,0,-1))}}}.  The basis for eigenspace for λ = -1 is then {{{ (matrix(3,1,1,0,-1)) }}}, and this is its eigenvector.