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Problem 6: Obtain row-reduction of $A$ and reduced row reduction of $A$.

 Problem 6: [Gaussian elimination] Let  \[\left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 &5 \\ 1 & 1 & 5 & 2 & 7 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right]\] be a given matrix. Obtain row-reduction of $A$ and reduced row reduction of $A$.

Solution:

We apply elementary row operations as follows to reduce the system to row echelon form.

\[\begin{align*} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 1 & 1 & 5 & 2 & 7 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right] \xrightarrow{R_2\rightarrow R_2-R_1} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 1 & 2 & 8 & 4 & 12 \\ \end{array}\right]. \end{align*}\] \[\xrightarrow{R3\rightarrow R_3-R_1} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 0 & 2 & 6 & 3 & 7 \\ \end{array}\right] \xrightarrow{R3\rightarrow R_3-2R_2} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 1 & 5 \\ 0 & 1 & 3 & 1 & 2 \\ 0 & 0 & 0 & 1 & 3 \\ \end{array}\right]\]

Row reduction form

\[\xrightarrow{R2\rightarrow R_2-R_1,\;R1\rightarrow R_1-R_3} \left[\begin{array}{rrrr|r} 1 & 0 & 2 & 0 & 2 \\ 0 & 1 & 3 & 0 & -1 \\ 0 & 0 & 0 & 1 & 3 \\ \end{array}\right]\]

Reduced Row reduction form

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