Question: Let A, B M n ( R ). (a) Show that A is nonsingular and A 1 is nonnegative if and only if whenever

Let A, B ∈ M_n(R). (a) Show that A is nonsingular and A⁻¹ is nonnegative if and only if whenever x, y ∈ Rⁿ and Ax ≥ Ay, then x ≥ y. (b) A is said to be a monotone matrix if it satisfies either of the equivalent conditions in (a). If A and B are monotone matrices, show that AB is a monotone matrix. (c) Explain why every M-matrix is a monotone matrix.

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