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Assignment 3 - Assignment#3 Section 1.5(pg 45 1b Suppose...

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Assignment #3 Section 1.5 (pg 45) 1b) Suppose det(B) is zero. . . . Thus B is a singular matrix. Therefore if B is a non singular matrix then det(B) is not zero. 1d) Suppose x is a real number other than zero. Then x has a multiplicative inverse because x(1/x) = 1. Suppose x has another multiplicative inverse z. . . . Then P, where P is some proposition. . . . Then ~P. Therefore P and ~P, which is a contradiction. We conclude that x has only one multiplicative inverse. 1f) Part 1. Suppose A is compact. . . . Therefore A is closed and bounded. Part 2. Suppose A is closed and bounded. . . . Therefore A is compact. 2) If A and B are invertible matrices, then AB is invertible. a) Suppose AB is not invertible . . . Thus, A is not invertible or B is not invertible. Therefore is A is not invertible or B is not invertible, AB is not invertible. b) Suppose A is not invertible or B is not invertible. . . . Thus AB is not invertible. Therefore if AB is invertible then A and B are both invertible. c) Suppose both A and B are invertible, and AB is not. . .
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. Then G, where G is some proposition. . . . Then ~G. Hence G and ~G, which is a contradiction. Therefore if A and B are invertible, then AB is invertible. d) Suppose AB is invertible, and at least one of A or B is not invertible. . . . Then G . . . Then ~G. Hence G and ~g. Thus if AB is invertible, then both A and B are invertible. e)Part 1. Assume A and B are invertible . . . Therefore AB is invertible. Part 2. Assume AB is invertible. . . . Then A and B are invertible. We conclude that A and B are invertible if and only if AB is invertible. 3c) Prove by contraposition: x^2 is not divisible by 4 then x is odd. Pf: Suppose the integer x is not odd. Then, x is even, so there exists an integer j such that x=2j. Therefore, x^2=(2j)(2j)=4j, or that x^2 is divisible by 4.
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