If A and B are different matrices satisfying A3 = B3 and A2B = B2A, then

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Introduction to Determinants

Question

If A and B are different matrices satisfying A³ = B³ and A²B = B²A, then

det (A – B) must be zero.

det (A² + B²) as well as det (A – B) must be zero.

det (A² + B²) must be zero.

At least one of det (A² + B²) or det (A – B) must be zero.

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A³ = B³   ....(1)
and   A²B = B²A   ....(2)
(1) – (2) gives,   A³ – A²B = B³ – B²A
A²(A – B) = – B²(A – B)   ⇒   (A² + B²)(A – B) = 0
det (A² + B²)(A – B) = 0
det (A² + B²) · det (A – B) = 0