# Method for finding the variable matrix of $Ax = b$

As the title stated, what is the most efficient method for finding the variable matrix?

For example, in the linear system $Ax = b$, given $$A = \begin{pmatrix} 1 & 1& 5 \\ 2& 3& 4 \\ 0& 1& -1\\ 1 & 1& 2 \\ \end{pmatrix}$$ and $$b = \begin{pmatrix} 16 \\ 13 \\ -4\\ 7 \\ \end{pmatrix}$$ what would then be the variable matrix $x$?

For my attempt, i've thought of reducing $(A|b)$ into its reduced row-echelon form:

I've got the $I_{4}$ instead, which probably doesn't tell me much.

Please advise, thanks.

## 1 Answer

Why don't you think it tells you much? If you've reduced the equation to $$I_{4\times 4} x = \tilde b$$ for some new right-hand side $\tilde b$, what is $x$?

• Does it mean that $x = (0, 0, 0)$? However, apparently i've got the answer that $x = (2, -1, 3)$ as the solution to the linear system. How does one derive that variable matrix? – idolo Oct 28 '17 at 10:25