If `A=[2-1 1-1 2-1 1-1 2]` . Verify that `A^3-6A^2+9A-4I=O` and hence find `A^(-1)` .

Explanation

Given, `A=[[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]]`

Calculating `A^2`

`A^2=A.A`

`A^2=[[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]][[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]]`

` " " " " =[[2(2)+(-1)(-1)+1(1),2(-1)+(-1)2+1(-1),2(1)+(-1)(-1)+1(2)],[-1(2)+2(-1)+(-1)(1),-1(-1)+2(2)+(-1)(-1),-1(1)+2(-1)+(-1)2],[1(2)+(-1)(-1)+2(1),1(-1)+(-1)(2)+2(-1),1(1)+(-1)(-1)+2(2)]]`

` " " " " =[[4+1+1,-2-2-1,2+1+2],[-2-2-1,1+4+1,-1-2-2],[2+1+2,-1-2-2,1+1+4]]`

`A^2=[[6,-5,5],[-5,6,-5],[5,-5,6]]`


Calculating `A^3`

`A^3=A^2.A`

`A^3=[[6,-5,5],[-5,6,-5],[5,-5,6]].[[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]]`

` " " " " =[[6(2)+(-5)(-1)+5(1),6(-1)+(-5)(2)+5(-1),6(1)+(-5)(-1)+5(2)],[-5(2)+6(-1)+(-5)(1),-5(-1)+6(2)+(-5)(-1),-5(1)+6(-1)+(-5)(2)],[5(2)+(-5)(-1)+6(1),5(-1)+(-5)2+6(-1),5(1)+(-5)(-1)+6(2)]]`

` " " " " =[[12+5+5,-6-10-5,6+5+10],[-10-6-5,5+12+5,-5-6-10],[10+5+6,-5-10-6,5+5+12]]`

`A^3=[[22,-21,21],[-21,22,-21],[21,-21,22]]`

Now, putting values in `A^3-6A^2+9A-4I`

` " " " " =[[22,-21,21],[-21,22,-21],[21,-21,22]]-6[[6,-5,5],[-5,6,-5],[5,-5,6]]+9[[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]]-4[[1,0,0],[0,1,0],[0,0,1]]`

` " " " " =[[22,-21,21],[-21,22,-21],[21,-21,22]]-[[36,-30,30],[-30,36,-30],[30,-30,36]]+[[18,-9, 9],[-9 ,18,-9],[ 9, -9, 18]]-[[4,0,0],[0,4,0],[0,0,4]]`

` " " " " =[[22-36+18+4,-21-(-30)+(-9)+0,21-30+9+0],[-21-(-30)+(-9)60,22-36+18+4,-22-(-30)+(-9)+0],[21-30+9+0,-21-(-30)+(-9)+0,22-36+18+4]]`

` " " " " =[[0,0,0],[0,0,0],[0,0,0]]=0`

Hence Proved


Now calculating `A^(-1)` using

`A^3-6A^2+9A-4I=0`
Post multiplying by `A^(-1)` on both sides

`(A^3-6A^2+9A-4I)A^(-1)=0.A^(-1)`

`A^3.A^(-1)-6A^2.A^(-1)+9A.A^(-1)-4IA^(-1)=0`

`A^2.A.A^(-1)-6A.A.A^(-1)+9A.A^(-1)-4IA^(-1)=0`

`A^2I-6AI+9I-4IA^(-1)=0`

`A^2-6A+9I-4A^(-1)=0`

`4A^(-1)=A^2-6A+9I`

`A^(-1)=1/4(A^2-6A+9I)`

Putting value

`A^(-1)=1/4([[6,-5,5],[-5,6,-5],[5,-5,6]]-6[[2,-1, 1],[-1 ,2,-1],[ 1, -1, 2]]+9[[1,0,0],[0,1,0],[0,0,1]])`

` " " " " =A^(-1)=1/4([[6,-5,5],[-5,6,-5],[5,-5,6]]-[[(6)2,(6)(-1), (6)1],[-1(6) ,(6)2,-1(6)],[ (6)1, -1(6),(6) 2]]+[[(9)1,0,0],[0,(9)1,0],[0,0,(9)1]])`

`A^(-1)=1/4([[6,-5,5],[-5,6,-5],[5,-5,6]]-[[12,-6, 6],[-6 ,12,-6],[ 6, -6, 12]]+[[9,0,0],[0,9,0],[0,0,9]])`

`A^(-1)=1/4[[6-12+9,-5+6+0,5-6+0],[-5+6+0,6-12+9,-5+6+0],[5-6+0,-5+6+0,6-12+9]]`

`A^(-1)=1/4[[3,1,-1],[1,3,1],[-1,1,3]]`