Question

If A and B are square matrices of order n, then prove that A and B will commute if and only if A − λI and B − λI commute for every scalar λ​

Answer 1

Step-by-step explanation:

$Given, (A−λI)(B−λI)=(B−λI)(A−λI)</p><p></p><p>$

$⇒AB−λA−λB+λ2=BA−λA−λB+λ2</p><p></p><p>$

$⇒AB=BA</p><p></p><p>$

Answer 2

Given, (A−λI)(B−λI)=(B−λI)(A−λI)

⇒AB−λA−λB+λ  

2

=BA−λA−λB+λ  

2

 

⇒AB=BA