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I am given

\begin{bmatrix} 1 & 1 & 1 & 1 \\[0.3em] 1 & 2 & 4 & 2 \\[0.3em] 2 & 0 & -4 & 1 \end{bmatrix}

I know that I need to get it in reduced row echelon form so I first do $-R_1 + R_2$ to get.

\begin{bmatrix} 1 & 1 & 1 & 1 \\[0.3em] 0 & 1 & 3 & 1 \\[0.3em] 2 & 0 & -4 & 1 \end{bmatrix}

Now I know that I need to get the row 3 2 into a zero so I do $-2R+1 + R_3$

\begin{bmatrix} 1 & 1 & 1 & 1 \\[0.3em] 0 & 1 & 3 & 1 \\[0.3em] 0 & -2 & -6 & -1 \end{bmatrix}

Now I need the row 3 column 2 to be zero. so $2R_2 + R_3$ \begin{bmatrix} 1 & 1 & 1 & 1 \\[0.3em] 0 & 1 & 3 & 1 \\[0.3em] 0 & 0 & 0 & 1 \end{bmatrix}

This is inconsistent, was the step I did wrong or the system wrong? What went wrong? How do I find the rank and nullity on something that is inconsistent like this?

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    $\begingroup$ What do you mean $2x3$? $\endgroup$ – Amzoti Sep 22 '13 at 20:31
  • $\begingroup$ @Amzoti When talking about a matrix you call them by their rows and columns, and it is always rows and then columns so I mean 2 rows by 3 columns. Obviously that is a typo. Are you confused? $\endgroup$ – Paul the Pirate Sep 22 '13 at 20:40
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We perform the row operations: \begin{align*} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 2 \\ 2 & 0 & -4 & 1 \\ \end{bmatrix} & \sim_{R_2 \gets R_2-R_1} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 3 & 1 \\ 2 & 0 & -4 & 1 \\ \end{bmatrix} \\ & \sim_{R_3 \gets R_3-2R_1} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 3 & 1 \\ 0 & -2 & -6 & -1 \\ \end{bmatrix} \\ & \sim_{R_3 \gets R_3+2R_2} \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 3 & 1 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \\ \end{align*} to obtain the row echelon form (in agreement with the OP's work).

It doesn't really make sense to talk about consistency here; it's just a matrix, not a system of equations.

We've shown that the row echelon form has $3$ leading $1$'s and thus the matrix has rank $3$, and thus the Rank-Nullity Theorem implies it has nullity $1$.

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  • $\begingroup$ Oh ok I was more concerned with what to do when I run into a situation like that representing a system of equations. Does it always mean it is inconsistent and do I really need to get reduced row echelon form to see that I have a rank of 3? I mean even what i have isn't reduced and I got the right answer. $\endgroup$ – Paul the Pirate Sep 22 '13 at 20:45
  • $\begingroup$ If it were an augmented matrix, then it would be inconsistent. But this doesn't stop us finding the row echelon form and reduced row echelon form in the same way. $\endgroup$ – Rebecca J. Stones Sep 22 '13 at 20:47
  • $\begingroup$ But do I need the reduced row echelon form always to find the rank? $\endgroup$ – Paul the Pirate Sep 22 '13 at 20:49
  • $\begingroup$ Don't need the reduced row echelon form, just the row echelon form. $\endgroup$ – Rebecca J. Stones Sep 22 '13 at 20:52
  • $\begingroup$ Alright thanks. $\endgroup$ – Paul the Pirate Sep 22 '13 at 20:53

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