Let A be a 3 3 real matrix such that A^2(A-2 I)-4( ~A - I )= O , where I and O are the identity and null matrices, respectively. If A^5= A^2+ A+ I , where , and are real constants, then + + is equal to:

Step-by-step solution

aligned & A^3-2 A^2-4 A+4 I=0 & A^3=2 A^2+4 A-4 I & A^4=2 A^3+4 A^2-4 A & =2 (2 A^2+4 A-4 I )+4 A^2-4 A & A^4=8 A^2+4 A-8 I & A^5=8 A^3+4 A^2-8 A & =8 (2 A^2+4 A-4 I )+4 A^2-8 A & A^5=20 A^2+24 A-32 I & =20, =24, =-32 & + + =12 aligned