Explanation a) Since all entries of all matrices are positive, it is clear that AC ≥ O and BD ≥ O . To see that AC ≥ BD, let A = (a ij ), B = (b ij ), C = (c ij ), and D = (d ij ) . Then each a ij ≥ b ij and each c ij ≥ d ij . So (AC) ij = ∑ k = 1 n a ik c kj ≥ ∑ k = 1 n b ik c kj ≥ ∑ k = 1 n b ik d kj = (BD) ij . So every entry of AC is greater than or equal to the corresponding entry of BD, and thus AC ≥ BD . b) Let x T = [x 1 x 2

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Author: David Poole

Publisher: Cengage Learning

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

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Chapter 3.7, Problem 36EQ

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Let A, B, C, and D be n×n matrices and x and y vectors in Rn. Prove the following inequalities: (a) If A≥B≥O and C≥D≥O, then AC≥BD≥O.(b) If A > B and x≥0, x ≠ 0, then Ax > Bx.

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Chapter 3.7, Problem 35EQ

Chapter 3.7, Problem 37EQ

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Let A, B, C, and D be n × n matrices and x and y vectors in R n . Prove the following inequalities: (a) If A ≥ B ≥ O and C ≥ D ≥ O, then AC ≥ BD ≥ O . (b) If A > B and x ≥ 0, x ≠ 0, then Ax > Bx .

Solution Summary: The author explains that the following inequalities hold true: ACge O and BD.

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Let A, B, C, and D be n×n matrices and x and y vectors in Rn. Prove the following inequalities: (a) If A≥B≥O and C≥D≥O, then AC≥BD≥O.(b) If A > B and x≥0, x ≠ 0, then Ax > Bx.

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Chapter 3 Solutions

Let A, B, C, and D be n × n matrices and x and y vectors in R n . Prove the following inequalities: (a) If A ≥ B ≥ O and C ≥ D ≥ O, then AC ≥ BD ≥ O . (b) If A &gt; B and x ≥ 0, x ≠ 0, then Ax &gt; Bx .

Solution Summary: The author explains that the following inequalities hold true: ACge O and BD.

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Let A, B, C, and D be n×n matrices and x and y vectors in Rn. Prove the following inequalities: (a) If A≥B≥O and C≥D≥O, then AC≥BD≥O.(b) If A > B and x≥0, x ≠ 0, then Ax > Bx.

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Chapter 3 Solutions

Let A, B, C, and D be n × n matrices and x and y vectors in R n . Prove the following inequalities: (a) If A ≥ B ≥ O and C ≥ D ≥ O, then AC ≥ BD ≥ O . (b) If A > B and x ≥ 0, x ≠ 0, then Ax > Bx .