Question

If A is a non singular matrix satisfying $A=AB-BA$, then which one of the following holds true

• A. $det.B=0$
• B. $B=0$
• C. $det.A=1$
• D. $det(B+I)=det(B-I)$

Answer 1

The correct option is D $det(B+I)=det(B-I)$

$A=AB-BA$
$AI+BA=AB$
$A(I+B)A=AB$
$|I+B|\left|A\right|=\left|A\right|\left|B\right|$
$\left|B\right|=|I+B|⟶1$
$A=AB-BA$
$BA=AB-A$
$BA=A(B-I)$
$\left|B\right|\left|A\right|=\left|A\right||B-I|$
$\left|B\right|=|B-I|⟶2$
From equation 1 and 2
$|B-I|=|B+I|$