The Product and the Determinant of Matrices with Entries in a Field

Let us considerK and letA be a matrix overK. The functor−A yields a matrix overK and is defined by: (Def. 2) len(−A) = lenA and width(−A) = widthA and for alli, j such that〈i, j〉 ∈ the indices of A holds(−A)◦ (i, j) =−(A◦ (i, j)). Let us consider K and letA, B be matrices over K. The functorA+B yields a matrix overK and is defined by: (Def. 3) len(A+B) = lenA and width(A+B) = widthA and for alli, j such that〈i, j〉 ∈ the indices of A holds(A+B)◦ (i, j) = (A◦ (i, j))+(B◦ (i, j)). The following propositions are true:

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