Question

If $A$ and $B$ are squares matrices such that $A^{2006}=O$ and $AB=A+B$, then $det\left(B\right)$ equals

• A. $0$
• B. $1$
• C. $-1$
• D. none of these

Answer 1

The correct option is A $0$

Given, $A^{2006}=O$
Also,$AB=A+B$
Multiplying by $A^{2005}$, we get
$A^{2006}B=A^{2006}+A^{2005}B$
$O=0+A^{2005}B$
or $A^{2005}B=O$
$det\left(A^{2005}B\right)=0$
$\Rightarrow det{A}^{2005}|B|=0$
$\Rightarrow |B|=0$