Let ( S = m Z : A ^ m ^2 + A ^ m =3 I - A ⁻⁶ ), where ( A = [ array cc 2 & -1 1 & 0 array ] ). Then ( n ( S ) ) is equal to ______.

Step-by-step solution

A= [ array cc 2 & -1 1 & 0 array ] Now finding characteristic equation aligned & | array cc 2- & -1 1 & - array |=0 & (2- )(- )-(-1)(1)=-2 + ^2+1=0 & ^2-2 +1=0 & ( -1)^2=0 & =1 aligned Since A satisfies (A-I)^2=0 aligned & A=I+N where & N=A-I & N= [ array ll 1 & -1 1 & -1 array ] & N^2=0 & A^m=(I+N)^m=I+m N & A^m A^m=(I+m N)(I+m N)=I+2 m N+m^2 N^2 aligned Since N^2=0 A^ m^2 =I+2 m N Now putting in given condition aligned & I+m^2 N+I+m N=3 I-A⁻⁶ & A⁻¹= [ array cc 0 & 1 -1 & 2 array ] & A⁻⁶= (A⁻¹ )^6=I+(-6) N aligned