Question

If $A$ and $B$ are square matrices of order $n$, then $A-\lambda I$ and $B-\lambda I$ commute for every scalar $\lambda$ only if

• A. $AB=BA$
• B. $AB+BA=O$
• C. $A=-B$
• D. None of these

Answer 1

Answer: (A)

$(A-\lambda I)(B-\lambda I)=(B-\lambda I)(A-\lambda I)$
$\Rightarrow AB-\lambda (A+B)I+{\lambda }^2{I}^2$
$=BA-\lambda (B+A)I+{\lambda }^2{I}^2$
$\Rightarrow AB=BA$