Given general solution of a matrix, how to reconstruct its RREF form?

Question

Here's the problem I am trying to solve:

I get that the second column would have an independent variable (as it would be a pivotal column) and the third and first columns would have nonpivotal columns (as they could be expressed as free variables).

But what's next? How to approach constructing R and k matrices?

Answer 1

From general solution, it is clear that $\begin{bmatrix}1\\2\\0\end{bmatrix}$ and $\begin{bmatrix}0\\5\\1\end{bmatrix}$ forms basis for null space of $A$ and $R$.

So can you construct a RREF matrix $R$ with this information.

Note that if $C_1, C_2, C_3$ are columns of $R$, then we have $C_1+2C_2=0_{3\times1}$ and $5C_2+C_3=0_{3\times1}$.

Check this after you have solved.

$R=\begin{bmatrix}1 & \frac{-1}{2} & \frac{5}{2} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

After this, $k=R\begin{bmatrix}0\\4\\0\end{bmatrix}$.