Maui1opj

2023-03-30

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix
$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$

pandurorc2x

To find all solutions of the equation $Ax=0$, where $A$ is row equivalent to the matrix $\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$, we need to find the parametric vector form of the solutions.
We can start by performing row reduction on the augmented matrix $\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$.
Applying row operations, we obtain the row-reduced form:
$\left[\begin{array}{cccc}1& 3& 0& -4\\ 0& 0& 0& 0\end{array}\right]$
The second row indicates that we have a free variable. Let's call it $t$.
Now, we can express the solutions in terms of the free variable $t$. Rewriting the row-reduced form as equations, we have:
$x+3y-4z=0$
$0=0$
We can solve the first equation for $x$:
$x=-3y+4z$
Expressing the solution in vector form, we have:
$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-3y+4z\\ y\\ z\end{array}\right]=y\left[\begin{array}{c}-3\\ 1\\ 0\end{array}\right]+z\left[\begin{array}{c}4\\ 0\\ 1\end{array}\right]$
Therefore, the solutions of the equation $Ax=0$ in parametric vector form are:
$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=y\left[\begin{array}{c}-3\\ 1\\ 0\end{array}\right]+z\left[\begin{array}{c}4\\ 0\\ 1\end{array}\right]$
where $y$ and $z$ are real numbers that serve as parameters.

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