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How would I reduce my matrix even further?

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Thomas
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Megan Reynolds
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  • The picture doesn't load. –  Sep 30 '13 at 11:12
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    You're already there, and your answer is correct. – DonAntonio Sep 30 '13 at 12:15
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    The only further simplification is multiplying each row by the inverse of its leftmost non zero coefficient. In this way, the final column gives you directly the (unique) solution of your system. – egreg Sep 30 '13 at 12:38

2 Answers2

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I usually do also pivot reduction: \begin{align} \large \begin{bmatrix} 2 & 2 & 1 & 2\\ -1 & 2 & -1 & -5\\ 1 & -3 & 2 & 8 \end{bmatrix} \xrightarrow{E_1(1/2)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ -1 & 2 & -1 & -5\\ 1 & -3 & 2 & 8 \end{bmatrix} \\ \large \xrightarrow{E_{21}(1)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ 0 & 3 & -1/2 & -4\\ 1 & -3 & 2 & 8 \end{bmatrix} \\ \large \xrightarrow{E_{31}(-1)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ 0 & 3 & -1/2 & -4\\ 0 & -4 & 3/2 & 7 \end{bmatrix} \\ \large \xrightarrow{E_{2}(1/3)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ 0 & 1 & -1/6 & -4/3\\ 0 & -4 & 3/2 & 7 \end{bmatrix} \\ \large \xrightarrow{E_{32}(4)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ 0 & 1 & -1/6 & -4/3\\ 0 & 0 & 5/6 & 5/3 \end{bmatrix} \\ \large \xrightarrow{E_{3}(6/5)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ 0 & 1 & -1/6 & -4/3\\ 0 & 0 & 1 & 2 \end{bmatrix} \\ \text{Backwards elimination starts}\\ \large \xrightarrow{E_{23}(1/6)} &\large \begin{bmatrix} 1 & 1 & 1/2 & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 2 \end{bmatrix} \\ \large \xrightarrow{E_{13}(-1/2)} &\large \begin{bmatrix} 1 & 1 & 0 & 0\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 2 \end{bmatrix} \\ \large \xrightarrow{E_{12}(-1)} &\large \begin{bmatrix} 1 & 0 & 0 & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 2 \end{bmatrix} \end{align} In this way you can read directly the solution in the last column: $$ x=1,\quad y=-1,\quad z=2. $$

Notation.

  1. $E_{i}(c)$ (with $c\ne0$) is “multiply the $i$-th row by $c$”.

  2. $E_{ij}(d)$ (with $i\ne j$) is “sum to the $i$-th row the $j$-th row multiplied by $d$”.

egreg
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You can't reduce it any further. It is already in RREF.

HINT: What are the conditions for RREF?

Don Larynx
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