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x1+x2+2x3=1x_{1}+x_{2}+2x_{3}=-1\newline x12x2+x3=5x_{1}-2x_{2}+x_{3}=-5 \newline3x1+x2+x3=33x_{1}+x_{2}+x_{3}=3 \newlineFind all solutions by using the Gaussian elimination & Gauss-Jordan Reduction.

Full solution

Q. x1+x2+2x3=1x_{1}+x_{2}+2x_{3}=-1\newline x12x2+x3=5x_{1}-2x_{2}+x_{3}=-5 \newline3x1+x2+x3=33x_{1}+x_{2}+x_{3}=3 \newlineFind all solutions by using the Gaussian elimination & Gauss-Jordan Reduction.
  1. Write Augmented Matrix: Write the augmented matrix for the system of equations.\newline[1amp;1amp;2amp;amp;11amp;2amp;1amp;amp;53amp;1amp;1amp;amp;3] \begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 1 & -2 & 1 & | & -5 \\ 3 & 1 & 1 & | & 3 \end{bmatrix}
  2. Leading 11 in First Row: Perform row operations to get a leading 11 in the first row, first column (R1R_1 is already set).\newlineNo changes needed for R1R_1.
  3. Eliminate Below Leading 11: Make the elements below the leading 11 in the first column zero, using R22 - R11 → R22 and R33 - 33R11 → R33.\newline[1amp;1amp;2amp;amp;10amp;3amp;1amp;amp;40amp;2amp;5amp;amp;6] \begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & -3 & -1 & | & -4 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}
  4. Leading 11 in Second Row: Get a leading 11 in the second row, second column by dividing R22 by 3-3.\newline[1amp;1amp;2amp;amp;10amp;1amp;1/3amp;amp;4/30amp;2amp;5amp;amp;6] \begin{bmatrix} 1 & 1 & 2 & | & -1 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & -2 & -5 & | & 6 \end{bmatrix}
  5. Eliminate Above and Below: Eliminate the entries above and below the leading 11 in the second column. Add R22 to R11 and add 22R22 to R33.\newline[1amp;0amp;7/3amp;amp;1/30amp;1amp;1/3amp;amp;4/30amp;0amp;7/3amp;amp;14/3] \begin{bmatrix} 1 & 0 & 7/3 & | & 1/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & -7/3 & | & 14/3 \end{bmatrix}
  6. Leading 11 in Third Row: Get a leading 11 in the third row, third column by dividing R33 by 7-7/33.\newline[1amp;0amp;7/3amp;amp;1/30amp;1amp;1/3amp;amp;4/30amp;0amp;1amp;amp;2] \begin{bmatrix} 1 & 0 & 7/3 & | & 1/3 \\ 0 & 1 & 1/3 & | & 4/3 \\ 0 & 0 & 1 & | & -2 \end{bmatrix}
  7. Eliminate Above Leading 11: Eliminate the entries above the leading 11 in the third column. Subtract (77/33)R33 from R11 and subtract (11/33)R33 from R22.\newline[1amp;0amp;0amp;amp;50amp;1amp;0amp;amp;50amp;0amp;1amp;amp;2] \begin{bmatrix} 1 & 0 & 0 & | & 5 \\ 0 & 1 & 0 & | & 5 \\ 0 & 0 & 1 & | & -2 \end{bmatrix}