Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

Question: Let A be the n x n matrix with 0^'s on the main diagonal, and 1's everywhere else. For an arbitrary vector $\vec{b}$ in $□^{n}$ , solve the linear system $A \vec{x} = \overset{⇀}{b}$ , expressing the components $x_{1}, . . . . . . ., x_{n}$ of $\vec{x}$ in terms of the components of $\overset{⇀}{b}$ . See Exercise 69 for the case n=3 .

The solution of the linear system $A \vec{x} = \overset{⇀}{b}$ is $x_{i} = \frac{b_{1} + . . . . . . + b_{n}}{n - 1} - b_{i}, i = 1, 2, . . . ., n$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the system.

If A is an n x m matrix with row vectors $\overset{⇀}{ω_{1}}, . . . . . . . . . . \overset{⇀}{ω_{n}}$ and $\overset{⇀}{x}$ is a vector in R^m then, .

$A \overset{⇀}{x} = [\begin{matrix} - \overset{⇀}{ω_{1}} - \\ . \\ . \\ . \\ - \overset{⇀}{ω_{n}} - \end{matrix}] \overset{⇀}{x} = [\begin{matrix} - \overset{⇀}{ω_{1}} . \overset{⇀}{x} - \\ . \\ . \\ . \\ - \overset{⇀}{ω_{n}} . \overset{⇀}{x} - \end{matrix}]$

Consider the linear system.

$[\begin{matrix} y + z = a \\ y + z = b \\ y + y = c \end{matrix}]$

The matrix form of the system is,

$[\begin{matrix} 0 & 1 & 1 & | & 1 \\ 1 & 0 & 1 & | & b \\ 1 & 1 & 0 & | & c \end{matrix}]$

The solution is, $x = \frac{b + c - a}{2}, y = \frac{a + c - b}{2}, z = \frac{a + b - c}{2}$ .

Step 2: Compute the system

Consider the linear system, $x_{1}, . . . . . . ., x_{n}$ of $\overset{⇀}{x}$ $\overset{⇀}{x}$ in terms of the components of $\overset{⇀}{b}$ .

$[\begin{matrix} x_{1} + x_{2} + . . . . . . . + x_{n} = b_{1} \\ x_{1} + x_{2} + . . . . . . . + x_{n} = b_{1} \\ . \\ ; \\ x_{1} + x_{2} + . . . . . . . + x_{n} = b_{n - 1} \\ x_{1} + x_{2} + . . . . . . . + x_{n} = b_{n} \end{matrix}]$

The solution, $x_{i} = \frac{b_{1} + . . . . . . + b_{n}}{n - 1} - b_{i}$ .

Where, $i = 1, 2, . . . ., n$

Hence, $x_{i} = \frac{b_{1} + . . . . . . + b_{n}}{n - 1} - b_{i}, i = 1, 2, . . . ., n$ is the solution of the linear system $A \vec{x} = \vec{b} .$

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Linear Algebra and its Applications

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Step by Step Solution

Step 1: Consider the system.

Step 2: Compute the system

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Linear Algebra and its Applications

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Short Answer

Question: Let A be the n x n matrix with 0's on the main diagonal, and 1's everywhere else. For an arbitrary vector b→ in □n, solve the linear system A x→=b⇀ , expressing the components x1,.......,xn of x→ in terms of the components of b⇀ . See Exercise 69 for the case n=3 .

Step by Step Solution

Step 1: Consider the system.

Step 2: Compute the system

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