Question

Q7. (a) Find a basis for the eigenspace of the following matrix corresponding to the eigenvalue X= 2: 4 -1 6 2 16 2 -1 8 (b) Suppose that the vector z is an eigenvector of the matrix A corresponding to the eigenvalue 4. Let n be a positive integer. What is A"r equal to?

Answer 1

solution 6 let h -| Aa 6 6 - 2 2 -| 8 Now to find basis for the the eigenvalue 9 = 2 eigen space of For this we need to solue (A - 11) - V3 O rioo where 1 - 0 001 Now put A=2 we get 2 t 6 2 -1 2 1 6 6 O Now I RI we get Nit 01 2 -1 na w 2 3 R2# ZRI. 2 Riglo R3 -2R1 0 0 0 o 2 80 164 1/² R3 II 3 o

Interchange Rz and R3 1 1 3 3/10 0 o o 86 01-18 Rita R2 0 3 O - 0 and Thus here Uz is free variable Vi+Uzz0 02=0 Now take 03=t 0,=-3t U1 -3t mo t - 13 Thus Basis for eigenspace is mo no solution © let a is an eigen vechor of the matrix A with corresponding eigen value a » Ax= px How ÅR A(AX) A(92) = 1 (Ar) Ax= A (2x) A²x = 2² 2

Next Åx = A (A) = A (2) > A3x = ²x 2²(Ax) = ₂(2x) Thus continuing in this way we get n Ax=da Thank you.