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  4. If A and B are square matrices of order n, then A-λ I and B-λ I commute for every scalar λ only if

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Q. 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

Matrices

Solution:

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