Question 5: (13 marks)\ Let

a,b,c,dinR

and

\\\\gamma =(a+d)^(2)-4(ad-bc)

and

M=([1,0,0],[0,a,b],[0,c,d])

.\ a. If

ac!=c,bc!=0

and

(a-1)(d-1)!=bc

, describe the eigenspace of

M

\ corresponding to 1\ b. If

ac!=c,bc!=0

and

(a-1)(d-1)=bc

, show that all eigenvectors corresponding\ to 1 can be exporessed in the form

(1,0,0)t+(0,(b)/(1-a),1)s

where

s,tinC

.\ (5 marks)\ c. If

\\\\gamma >0

and

a+d-2!=\\\\sqrt(\\\\gamma )

, why is M diagonalizable? Explain your answer.

Let a,b,c,dR and =(a+d)24(adbc) and M=1000ac0bd. a. If ac=c,bc=0 and (a1)(d1)=bc, describe the eigenspace of M corresponding to 1 (4 marks) b. If ac=c,bc=0 and (a1)(d1)=bc, show that all eigenvectors corresponding to 1 can be exporessed in the form (1,0,0)t+(0,1ab,1)s where s,tC. (5 marks) c. If >0 and a+d2=, why is M diagonalizable? Explain your answer. (4 marks)