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Let matrices A,B∈Mn×n(R). Show that if A and B are each similar to some diagonal matrix,...

Let matrices A,B∈Mn×n(R). Show that if A and B are each similar to some diagonal matrix, and also have the same eigenvectors (but not necessarily the same eigenvalues), then  AB=BA.

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here the matrix have the same eign vectors but no necessarily the same eign values so we use the similaity reltaion , noting that both the matrix are diagonalizable , but there diagonal forms are different , since there eign values are different , but the diagonal forms commute , so we simply use these these facts , here I am giving the solution below,

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