Explanation Approach: Rational expression, a quotient obtained by division of two polynomials in form of p ( x , y ) q ( x , y ) where p ( x , y ) and q ( x , y ) are polynomials in such a way that the variables x , y do not assume values such that q ( x , y ) = 0 . If a b and c d are rational, then addition or subtraction of rational formed by multiplying them by appropriate expressions and equating the denominators using LCD and perform the operation on the numerators: a b + c d = a b ⋅ d d + c d ⋅ b b = a d b d + c b b d = a d + c b b d And,

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Author: Jerome E. Kaufmann, Karen L. Schwitters

Publisher: Cengage Learning

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Chapter 4.6, Problem 27PS

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1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.

Question 3 over a field K. In this question, MË(K) denotes the set of n × n matrices (a) Suppose that A Є Mn(K) is an invertible matrix. Is it always true that A is equivalent to A-¹? Justify your answer. (b) Let B be given by 8 B = 0 7 7 0 -7 7 Working over the field F2 with 2 elements, compute the rank of B as an element of M2(F2). (c) Let 1 C -1 1 [4] [6] and consider C as an element of M3(Q). Determine the minimal polynomial mc(x) and hence, or otherwise, show that C can not be diagonalised. [7] (d) Show that C in (c) considered as an element of M3(R) can be diagonalised. Write down all the eigenvalues. Show your working. [8]

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Chapter 4 Solutions

Intermediate Algebra

Ch. 4.1 - For Problems 1-10, answer true or false. When a...Ch. 4.1 - For Problems 1-10, answer true or false. A...Ch. 4.1 - For Problems 1-10, answer true or false. 3 is a...Ch. 4.1 - For Problems 1-10, answer true or false. A...Ch. 4.1 - For Problems 1-10, answer true or false. The...Ch. 4.1 - For Problems 1-10, answer true or false. The...Ch. 4.1 - Prob. 7CQ Ch. 4.1 - Prob. 8CQ Ch. 4.1 - Prob. 9CQ Ch. 4.1 - Prob. 10CQ

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