# Help with simplifying the second row of a $4 \times 4$ matrix to $(1, 0, 0 ,0)$

I wonder how you would approch the problem of simplifying the second row of the matrix: $$\left( \begin{array}{cccc} 3 & 1 & 8 & 1 \\ -1 & -3 & 0 & 2 \\ 3 & -1 & 5 & 3 \\ 4 & 2 & 10 & 12 \end{array} \right)$$

The second row should be $(1, 0, 0 ,0)$, What strategy are you using when approaching this kind of problem?

I cannot find any mutiples of rows and columns that gets me closer to a row two of $(1, 0, 0 ,0)$.

Thank you kindly for your help!

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What do you mean by "simplifying the second row"? Are you trying to row-reduce the matrix? – Dennis Gulko Mar 7 '13 at 12:16
And why does the second row has to be $\,(1,0,0,0)\,$? Where does this requirement come from? – DonAntonio Mar 7 '13 at 12:20
This is from an excercise in my mathbook, they said that I should simplify the second row to (1, 0 ,0 ,0) but I could not see how to do that. – Lukas Arvidsson Mar 7 '13 at 14:07

Begin by multiplying column $1$ (denoted $C_1$) by $(-1)$ to get a $1$ in position $(2,1)$. Then replace $C_2$ by $C_2-3C_1$, then replace $C_4$ by $C_4+2C_1$. Here $C_k$ means column $k$.