A theorem of linear algebra states that if A and B are invertible matrices, then the product AB is invertible. As in Exercise 1, outline
(a) a direct proof of the theorem.
(b) a direct proof of the converse of the theorem.
Reference:
Analyze the logical form of each of the following statements and construct just the outline of a proof. Since the statements may contain terms with which you are not familiar, you should not (and perhaps could not) provide any details of the proof.
(a) Outline a direct proof that if (G, *) is a cyclic group, then (G, *) is abelian.
(b) Outline a direct proof that if B is a nonsingular matrix, then the determinant of B is not zero.
(c) Suppose A, B, and C are sets. Outline a direct proof that if A is a subset of B and B is a subset of C, then A is a subset of C.
(d) Outline a direct proof that if the maximum value of the differentiable function on the closed interval [a, b] occurs at then either
(e) Outline a direct proof that if A is a diagonal matrix, then A is invertible whenever all its diagonal entries are nonzero.
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