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if A and B are squares matrices such tha...

if `A and B` are squares matrices such that `A^(2006)=O and A B=A+B , then ,"det"(B)` equals a.`0` b. `1` c. `-1` d. none of these

A

`-1`

B

`0`

C

`1`

D

None of these

Text Solution

AI Generated Solution

To solve the problem, we need to analyze the given conditions and derive the determinant of matrix B step by step. ### Step-by-Step Solution: 1. **Given Information**: We know that \( A^{2006} = O \) (where O is the zero matrix) and \( AB = A + B \). 2. **Understanding \( A^{2006} = O \)**: Since \( A^{2006} = O \), it implies that matrix \( A \) is a nilpotent matrix. A property of nilpotent matrices is that their determinant is zero. Therefore, we can conclude: \[ ...
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