If A and B are square matrices of order 3 such that `A^(3)=8 B^(3)=8I` and det. `(AB-A-2B+2I) ne 0`, then identify the correct statement(s), where `I` is identity matrix of order 3. (A) `A^(2)+2A+4I=O` (B) `A^(2)+2A+4I neO` (C) `B^(2)+B+I=O` (D) `B^(2)+B+I ne O`

Options

A. `A^(2)+2A+4I=O`
B. `A^(2)+2A+4I neO`
C. `B^(2)+B+I=O`
D. `B^(2)+B+I ne O`

Answer

A^(2)+2A+4I=O` and `B^(2)+B+I=O`

Explanation

`A^(3)=8I`
`implies (A-2I) (A^(2)+2A+4I)=O`
Also, `B^(3)=I`
`implies (B-I)(B^(2)+B+I)=0`
Now, det. `(AB-A-2B+2I) ne 0`
`implies` det. `((A-2I)(B-I)) ne 0`
So, `B-I` and `A-2I` are non-singular matrices.
`implies A^(2)+2A+4I=O` and `B^(2)+B+I=O`