{:[x_(1)+2x_(2)+3x_(3)],[x_(1)+(2)(2)+3(3)],[x_(1)+5=6]:} An mxn matnix A is quadx_(1)+5=6 In recuad row ecchelon form if it satisfies quadx_(1)=6 the follouinn annorties: {:[x_(1)+x_(2)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} a) Find all solutions by unang the Gausion elimination 2 Gaurs Jordan Reduction

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{:[x_(1)+2x_(2)+3x_(3)],[x_(1)+(2)(2)+3(3)],[x_(1)+5=6]:} An mxn matnix  A is  quadx_(1)+5=6 In recuad row ecchelon form if it satisfies  quadx_(1)=6 the follouinn annorties:  {:[x_(1)+x_(2)+2x_(3)=-1],[x_(1)-2x_(2)+x_(3)=-5],[3x_(1)+x_(2)+x_(3)=3]:} a) Find all solutions by unang the Gausion elimination 2 Gaurs Jordan Reduction

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Set up augmented matrix:  Set up the augmented matrix for the system of equations: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 1 & -2 & 1 & | & -5 \ 3 & 1 & 1 & | & 3 \end{bmatrix} $$Perform Gaussian elimination:  Perform the first step of Gaussian elimination: Make the first element of the first column a 1 (already done), and use it to zero out the rest of the first column. - Subtract the first row from the second row: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & -3 & -1 & | & -4 \ 3 & 1 & 1 & | & 3 \end{bmatrix} $$ - Subtract 3 times the first row from the third row: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & -3 & -1 & | & -4 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$Normalize second row:  Normalize the second row by dividing by -3: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & -2 & -5 & | & 6 \end{bmatrix} $$ - Use the second row to zero out the rest of the second column: - Add 2 times the second row to the third row: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & -13/3 & | & 22/3 \end{bmatrix} $$Use second row:  Normalize the third row by multiplying by -3/13: $$ \begin{bmatrix} 1 & 1 & 2 & | & -1 \ 0 & 1 & 1/3 & | & 4/3 \ 0 & 0 & 1 & | & -2 \end{bmatrix} $$ - Use the third row to zero out the rest of the third column: - Subtract 2 times the third row from the first row and subtract 1/3 times the third row from the second row: $$ \begin{bmatrix} 1 & 1 & 0 & | & 3 \ 0 & 1 & 0 & | & 2 \ 0 & 0 & 1 & | & -2 \end{bmatrix} $$Normalize third row:  Use the second row to zero out the first column: - Subtract the second row from the first row: $$ \begin{bmatrix} 1 & 0 & 0 & | & 1 \ 0 & 1 & 0 & | & 2 \ 0 & 0 & 1 & | & -2 \end{bmatrix} $$ This is the reduced row echelon form, showing that each variable has a unique solution.

$x_{1}+x_{2}+2x_{3}=-1$ $\newline$ $x_{1}-2x_{2}+x_{3}=-5$ $\newline$ $3x_{1}+x_{2}+x_{3}=3$ $\newline$ Find all solutions by using the Gaussian elimination & Gauss Jordan Reduction

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x1+x2+2x3=−1x_{1}+x_{2}+2x_{3}=-1x1​+x2​+2x3​=−1\newline x1−2x2+x3=−5 x_{1}-2x_{2}+x_{3}=-5x1​−2x2​+x3​=−5\newline 3x1+x2+x3=3 3x_{1}+x_{2}+x_{3}=33x1​+x2​+x3​=3\newline Find all solutions by using the Gaussian elimination & Gauss Jordan Reduction

$x_{1}+x_{2}+2x_{3}=-1$ $\newline$ $x_{1}-2x_{2}+x_{3}=-5$ $\newline$ $3x_{1}+x_{2}+x_{3}=3$ $\newline$ Find all solutions by using the Gaussian elimination & Gauss Jordan Reduction