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Matrix elements are the components, or entries, of a matrix, typically considered in a certain basis.
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votes
Possible errata Landau and Lifshitz in $\S29$ Matrix elements of vectors in Quantum Mechanic...
I don't know what is wrong with your reasoning, since it seems to return the right result in the special case where $A_{yy}=A_{zz}$. That said, I never saw this approach, and I searched in Landau'book …