Question

If $A$ is non-singular matrix satisfying $AB-BA=A$, then

• A. $det(B+I)=det(B-I)$
• B. $det(B+I)=I$
• C. $det(B+I)=detA\times detB$
• D. $detA=detB$

Answer 1

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

$A$ is non-singular $\Rightarrow detA\ne 0$
Given: $AB-BA=A$
Hence $AB=A+BA=(I+B)A$
$\Rightarrow detA\times detB=det(I+B)\times detA$
$\Rightarrow detB=det(I+B)\cdots \left(1\right)$ (as $A$ is non-singular)

Also, we have $AB-A=BA$
$\Rightarrow A(B-I)=BA$
$\Rightarrow detA\times det(B-I)=detB\times detA$
$\Rightarrow det(B-I)=det\left(B\right)\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, we have: $det(B-I)=det(B+I)$