<p>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</p>
Step-by-Step Solution
Key Concept: Two matrices (A - λI) and (B - λI) commute for all scalars λ if and only if A and B themselves commute. This is because the commutativity condition reduces to AB = BA after expanding and using the fact that λ must work for every value.
<p><strong>Step 1:</strong> Expand the commutativity condition (A - λI)(B - λI) = (B - λI)(A - λI)</p><p><strong>Step 2:</strong> Left side: AB - λA - λB + λ²I</p><p>Right side: BA - λB - λA + λ²I</p><p><strong>Step 3:</strong> Subtracting both sides: AB - λA - λB + λ²I = BA - λB - λA + λ²I</p><p><strong>Step 4:</strong> Simplifying: AB = BA</p><p><strong>Step 5:</strong> Since this must hold for every scalar λ, the λ terms automatically cancel. The only constraint remaining is AB = BA.</p><p><strong>Step 6:</strong> Conversely, if AB = BA, then clearly (A - λI)(B - λI) = (B - λI)(A - λI) for all λ.</p><p>∴ Answer: A and B must commute (i.e., AB = BA)</p>
Correct Answer: A