Give the Characteristic Polynomial for A in fully factored form. Give a basis for the eigenspace for each eigenvalue of A. Work justifying your answers must be shown for credit to be 4 3 1 given. A = [4 3 1 0 3 4 0 1 3] Justify your answers. Fully factored Characteristic Polynomial: ___ 1st Eigenvalue: ____Basis for Eigenspace: ___ 2nd Eigenvalue: __ Basis for Eigenspace: ___ 3rd Eigenvalue: ___ Basis for Eigenspace: ____ Solution The characteristic equation of A is det(A- I3) = 0 or, 3 -102 +29 -20 = 0 or, (-5)(-4)(-1) = 0. The 1st eigenvalue of A is 1 = 5. The second eigenvalue of A is 2 = 4. The 3rd eigenvalue of A is 3 = 1 The eigenvector of A corresponding to 1 = 5 is the solution of the equation (A- 5I3)X = 0. The RREF of A- 5I3 is 1 0 -7 0 1 -2 0 0 0 If X = (x,y,z)T the above equation is equivalent to x-7z = 0 and y-2z = 0. Then X = (7z,2z,z)T = z(7,2,1)T. Hence the basis for the eigenspace of A corresponding to the eigenvalue 1 = 5 is {(7,2,1)T}. Similarly, the eigenvector of A corresponding to 2 = 4 is the solution of the equation (A- 4I3)X = 0. The RREF of A- 4I3 is 0 1 0 0 0 1 0 0 0 The equation (A- 4I3)X = 0 is equivalent to y = 0, z = 0. Hence X = (x,0,0)T = x(1,0,0)T. Hence the basis for the eigenspace of A corresponding to the eigenvalue 2 = 4 is {(1,0,0)T}. Also, the eigenvector of A corresponding to 3 = 1 is the solution of the equation (A- I3)X = 0. The RREF of A- I3 is 1 0 -5/3 0 1 2 0 0 0 The equation (A-I3)X=0 is equivalent to x-5z/3=0 and y+2z= 0.Let z= 3t. Then X=(5t, -6t,3t)T = t(5,-6,3)T. Hence the basis for the eigenspace of A corresponding to the eigenvalue 3 = 1 is {(5,- 6,3)T}. 1 0 -7 0 1 -2 0 0 0.