Question

matrix algebra

14. 0/3 points | Previous Answers HoltLinAlg2 6.1.067. Consider the matrix A 00-2-11 1 1 7 6 A=12041 Find the eigenvalues of A. (Enter your answers as a comma-separated list.) Find a basis for each eigenspace. 0 (smaller eigenvalue) (larger eigenvalue)

Answer 1

0 0 -2-1 -2 0 2 1 consider the characteristic polvnomial -λ 1 0-2 1-λ det(A-l)20 4-1 -2 0 therefore the eigen values are 1,1,2,2. now we have to find the eigen vector for the corresponding eigen value 1 0-1 0-2-1 11-1 761Xy 2 0 4-11x -2 0-2 1-1)(x 10 0 -1x, 0 0 x = -10 0 0 011x, 20 3x -20-2 0人 therefore the equation form is -x, = 0-24 = x,x, + x, = 0 free variable so, let x,s,x2 t Herex are 0 X, 10 0 0 0 now wehave to find the eigen vector for the corresponding eigen value 2 0-2 0-2-1)(X1 1 1-2 7 6 X2 2 0 4-2 1x -2 0 -2 1-2)x, A -21 -2 0 -2-1x 1-1 76x2 11 X2 :0 10 1-6 therefore the equation form is +-x,--x,--x, 4, Here X, X, are free variables so, let Х, 5,x-t

6 11/2 + t So, 1/2 11/2 Therefore the basis for the eigen space λ = 1 are , and for λ = 2 are ,,