Resolved: "Write the homogeneous system of linear equations in..."

College
Algebra
Linear algebra
Forms of linear equations

smileycellist2

Answered question

2021-03-04

Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number

c_{1}

{\begin{cases} x_{1} + x_{2} + x_{3} + x_{4} = 0 \\ - x_{1} + x_{2} - x_{3} + x_{4} = 0 \\ x_{1} + x_{2} - x_{3} - x_{4} = 0 \\ 3 x_{1} + x_{2} + x_{3} - x_{4} = 0 \end{cases}

X = (\begin{matrix} 1 \\ - 1 \\ - 1 \\ 1 \end{matrix})

Answer & Explanation

d2saint0

Skilled2021-03-05Added 89 answers

Lets write
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ - 1 & 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 3 & 1 & 1 & - 1 \end{matrix}] X = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]$
Matrix A , which has the dimensions $4 \times 4$ , is the matrix of ciefficients of the system , X , which has the dimensions $4 \times 1$ is the matrix of unknowns.
$A X = 0$
$[\begin{matrix} 1 & 1 & 1 & 1 \\ - 1 & 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 3 & 1 & 1 & - 1 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}]$
If X is a solution of SX=0 , then so is $c_{1}$ for any constant $c_{1}$ . compute the product AX:
$A X = [\begin{matrix} 1 & 1 & 1 & 1 \\ - 1 & 1 & - 1 & 1 \\ 1 & 1 & - 1 & - 1 \\ 3 & 1 & 1 & - 1 \end{matrix}] [\begin{matrix} 1 \\ - 1 \\ - 1 \\ 1 \end{matrix}]$
$= [\begin{matrix} 1 \cdot 1 + 1 \cdot (- 1) + 1 \cdot (- 1) + 1 \cdot 1 \\ - 1 \cdot 1 + 1 \cdot (- 1) + (- 1) \cdot (- 1) + 1 \cdot 1 \\ 1 \cdot 1 + 1 \cdot (- 1) + (- 1) \cdot (- 1) + (- 1) \cdot 1 \\ 3 \cdot 1 + 1 \cdot (- 1) + 1 \cdot (- 1) + (- 1) \cdot 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}]$

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