a.) Prove that if A 0 and B 0, then AB 0, assuming that...

Transcribed Image Text:

a.) Prove that if A ≥ 0 and B≥ 0, then AB ≥ 0, assuming that the product AB is defined. b.) (5 points) Prove that if A ≤ 0 and B ≥ 0, then AB ≤ 0, assuming that the product AB is defined. c.) (5 points) Prove that if A ≤ 0 and B≤ 0, then AB ≥ 0, assuming that the product AB is defined. d.) (5 points) Prove that if A ≥ 0 and B≥ C, then AB AC, assuming that all the products are defined. e.) (5 points) Prove that if A ≤ 0 and B≥ C, then AB ≤ AC, assuming that all the products are defined. f.) (3 points) Prove or disprove that A² ≥ 0 for all square matrices A. g.) (3 points) Prove or disprove that ATA ≥ 0 for all square matrices A. h.) (3 points) Prove or disprove that for all invertible square matrices A for which A ≥ 0, then A¹ ≥ 0. 1 a.) Prove that if A ≥ 0 and B≥ 0, then AB ≥ 0, assuming that the product AB is defined. b.) (5 points) Prove that if A ≤ 0 and B ≥ 0, then AB ≤ 0, assuming that the product AB is defined. c.) (5 points) Prove that if A ≤ 0 and B≤ 0, then AB ≥ 0, assuming that the product AB is defined. d.) (5 points) Prove that if A ≥ 0 and B≥ C, then AB AC, assuming that all the products are defined. e.) (5 points) Prove that if A ≤ 0 and B≥ C, then AB ≤ AC, assuming that all the products are defined. f.) (3 points) Prove or disprove that A² ≥ 0 for all square matrices A. g.) (3 points) Prove or disprove that ATA ≥ 0 for all square matrices A. h.) (3 points) Prove or disprove that for all invertible square matrices A for which A ≥ 0, then A¹ ≥ 0. 1

Answer rating: 100% (QA)

Properties of Products of Numbers and Matrices This response addresses the prompts questions regarding product inequalities and properties of specific ... View the full answer