0

I need clarification in understanding all solutions of $Ax=0$ in parametric vector form where the $A$ is row equivalent to the given matrix.

Matrix $A$ $$\begin{bmatrix} 1 & -4 & -2 & 0 & 3 & -5 \\ 0 & 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

The augmented matrix of $A$ $$ \left[ \begin{array}{cccccc|c} 1 & -4 & -2 & 0 & 3 & -5 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 1 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] $$

The Reduced Echelon Form $A$

$$ \left[ \begin{array}{cccccc|c} 1 & -4 & 0 & 0 & 0 & 9 & 0 \\ 0 & 0 & 1 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 1 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] $$

$x_1 = 4x_2 -9x_6 \\ x_2 is free \\ x_3 = 1x_6 \\ x_4 is free\\ x5 = 4x_6\\x_6 is free $

Which simplifies to

$x = $$\begin{bmatrix} x_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6 \end{bmatrix}$ = $x_2$$\begin{bmatrix} 4\\ 1\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$ + $x_6$$\begin{bmatrix} -9\\ 0\\ 1\\ 0\\ 4\\ 1 \end{bmatrix}$

Any help is appreciated! Please accept or reject the my solution to this equation and clarify mistakes (if any) that I have made.

Miroslav Glamuzina
  • 281
  • check your reduced matrix again and try to write out that each x1,...,x6. You have a 3x6 matrix with a rank of only 2, so you should end up with a 4-parameter family of solutions! –  Oct 06 '15 at 03:40
  • @Alex Hey there, I just saw and corrected my matrix, how does it look now? – Miroslav Glamuzina Oct 06 '15 at 04:26

1 Answers1

1

So this is a $3x6$ matrix with $rank=3$. Then the solution should have $n-r=3-6=3$ parameters! All of your work is correct except for the answer. In the answer you provided $x_4=0$ for any given $x_2,x_6$. In reality, $x_4$ is a parameter that can take on ANY value and you can see from your reduced matrix that $x_4$ is completely independent from the other variables! Your final solution should be:

$x = $$\begin{bmatrix} x_1\\ x_2\\ x_3\\ x_4\\ x_5\\ x_6 \end{bmatrix}$ = $x_2$$\begin{bmatrix} 4\\ 1\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$ + $x_4$$\begin{bmatrix} 0\\ 0\\ 0\\ 1\\ 0\\ 0 \end{bmatrix}$+ $x_6$$\begin{bmatrix} -9\\ 0\\ 1\\ 0\\ 4\\ 1 \end{bmatrix}$