Question

If $A,B$ are two square matrices of order $n$ and $A$ and $B$ commute $(K$ be a real number), then which of the following is correct

• A. $A-KI,B-KI$ are equal
• B. $A-KI,B-KI$ commute
• C. both $a$ and $b$ are correct
• D. $A-KI,B-KI$ do not commute

Answer 1

The correct option is B

$A-KI,B-KI$ commute

Given the matrices $A,B$ commute, then $AB=BA\cdots \left(1\right)$
Now, $(A-kI)(B-kI)=AB-k(A+B)+k^{2}I$ [Since $I^2=I$]
$=BA-k(B+A)+k^{2}I$
[Since $A+B=B+A\text{from}\left(1\right)$]$=(B-kI)(A-kI)$

So the matrices $(A-kI),(B-kI)$ commute.
Also $A-KI$ and $B-KI$ need not be equal.