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1 2 0 1 0 5 Given the matrix A is row equivalent to 0 0 1 -4 0 3 0 0 0 0 1 5 X = x2 + 000 000 X4 + " ☐☐☐☐☐☐ describe all solutions to Ax = 0. x6

1 2 0 1 0 5 Given the matrix A is row equivalent to 0 0 1 -4 0 3 0 0 0 0 1 5 X = x2 + 000 000 X4 + " ☐☐☐☐☐☐ describe all solutions to Ax = 0. x6

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given the matrix \( A \) is row equivalent to \[ \begin{bmatrix} 1 & -2 & 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & -4 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 5 \\ \end{bmatrix} \] describe all solutions to \( Ax = 0 \). The solution can be described as: \[ x = \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} = x_2 \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} + x_4 \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} + x_6 \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} \] where each matrix represents a column vector, and \( x_2, x_4, x_6 \) are free variables corresponding to the non-pivot columns in the reduced row echelon form of the matrix \( A \). Each of these columns represents a basis vector for the solution space of the homogeneous system \( Ax = 0 \).
Transcribed Image Text:Given the matrix \( A \) is row equivalent to \[ \begin{bmatrix} 1 & -2 & 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & -4 & 0 & 3 \\ 0 & 0 & 0 & 0 & 1 & 5 \\ \end{bmatrix} \] describe all solutions to \( Ax = 0 \). The solution can be described as: \[ x = \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} = x_2 \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} + x_4 \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} + x_6 \begin{bmatrix} \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \text{ } \\ \end{bmatrix} \] where each matrix represents a column vector, and \( x_2, x_4, x_6 \) are free variables corresponding to the non-pivot columns in the reduced row echelon form of the matrix \( A \). Each of these columns represents a basis vector for the solution space of the homogeneous system \( Ax = 0 \).
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