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Let ##A = (\alpha_{ij})## be an $n \times n# complex matrix. Define ##\hat## acting on ##A## as producing the matrix ##\hat{A} = (\alpha_{ij} I_n)##.

I don't understand what this is saying. Isn't ##I_n## the identity matrix, and therefore the product of it with any matrix results in the other matrix? If this is so, wouldn't ##\hat{A} = (\alpha_{ij} I_n) = (\alpha_{ij}) = A##?