Consider, `A=[(a,2,1),(0,b,0),(0,-3,c)]`, where a, b and c are the roots of the equation `x^(3)-3x^(2)+2x-1=0`. If matric B is such that `AB=BA, A+B-2I ne O` and `A^(2)-B^(2)=4I-4B`, then find the value of det. (B)

Explanation

Given that a, b and c are roots of the equation
`x^(3)-3x^(2)+2x-1=0`.
Let `x^(3)-3x^(2)+2x-1=(x-a) (x-b) (x-c)` (1)
Now, `A^(2)-B^(2)=4I-4B`
`:. A^(2)=B^(2)-4B+4I`
`implies A^(2)=(B-2I)^(2)`
`implies A^(2)-(B-2I)^(2)=O`
`implies (A+B-2I) (A-B+2I)=O" "("as "AB=BA)`
Now, `A+B-2I ne O`
`:. A-B+2I=O` [as otherwise `(A-B+2I)^(-1)` exists and `A+B-2I=O`]
`:. B=A+2I =[(a+2,2,1),(0,b+2,0),(0,-3,c+2)]`
`:. |B|=(a+2) (b+2) (c+2)` (2)
Putting `x=-2` in (1), we get
`-8-12-4-1=(-2-a) (-2-b) (-2-c)`
`:. 25=(a+2) (b+2) (c+2)`
`:. |B|=25`