Answered: Given the following augmented matrix…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter6: Matrices And Determinants

Section: Chapter Questions

Problem 10CC

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Question

Given the following augmented matrix representing a system of linear equations, find the solution.

Expert Solution

Step 1

Given:

AX=b where

$A = [\begin{matrix} 1 & 1 & 2 \\ 4 & 0 & 1 \\ 2 & 1 & 1 \end{matrix}]$ and $X = [\begin{matrix} x \\ y \\ z \end{matrix}]$ and $b = [\begin{matrix} 5 \\ 5 \\ 5 \end{matrix}]$

we write in the augmented matrix representation of given linear equation

Step 2

Determinant of A :

$|A| = |\begin{matrix} 1 & 1 & 2 \\ 4 & 0 & 1 \\ 2 & 1 & 1 \end{matrix}| = 1 |\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}| - 1 |\begin{matrix} 4 & 1 \\ 2 & 1 \end{matrix}| + 2 |\begin{matrix} 4 & 0 \\ 2 & 1 \end{matrix}| = 1 (0 - 1) - 1 (4 - 2) + 2 (4 - 0) = - 1 - 2 + 8 = 5 \neq 0$

Determinant of A is nonzero so given system has unique solution.

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