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If A and B are square matrices of order ...

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 idensity 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`

Text Solution

Verified by Experts

`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`
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