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Sagot :
Certainly! Let's use Gaussian elimination to solve the given system of equations:
[tex]\[ \begin{cases} -x + y + z = -2 \\ -x + 4y - 5z = -14 \\ 4x - 2y - 8z = 0 \end{cases} \][/tex]
We'll start by writing the augmented matrix of the system:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ -1 & 4 & -5 & | & -14 \\ 4 & -2 & -8 & | & 0 \end{pmatrix} \][/tex]
### Step 1: Forming Pivot in the First Column
We'll transform the matrix to get a leading 1 in the first row of the first column. For simplicity, let the first row remain the same, and we'll perform row operations to eliminate the variable [tex]\(x\)[/tex] from the second and third rows.
[tex]\[ R1: \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ -1 & 4 & -5 & | & -14 \\ 4 & -2 & -8 & | & 0 \end{pmatrix} \][/tex]
[tex]\[ R2 \gets R2 - R1 \][/tex]
Second row operation:
[tex]\[ -1 + 1 = 0, \quad 4 - 1 = 3, \quad -5 - 1 = -6, \quad -14 + 2 = -12 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 3 & -6 & | & -12 \\ 4 & -2 & -8 & | & 0 \end{pmatrix} \][/tex]
[tex]\[ R3 \gets R3 + 4 \cdot R1 \][/tex]
Third row operation:
[tex]\[ 4 + 4(-1) = 0, \quad -2 + 4(1) = 2, \quad -8 + 4(1) = 2, \quad 0 + 4(-2) = -8 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 3 & -6 & | & -12 \\ 0 & 2 & -8 & | & -8 \end{pmatrix} \][/tex]
### Step 2: Forming Pivot in the Second Column
Next, we'll make sure the second row has a leading one in the second column.
[tex]\[ R2 \gets \frac{1}{3} R2 \][/tex]
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 1 & -2 & | & -4 \\ 0 & 2 & -8 & | & -8 \end{pmatrix} \][/tex]
To eliminate the variable [tex]\(y\)[/tex] from the third row, we perform:
[tex]\[ R3 \gets R3 - 2 \times R2 \][/tex]
Third row operation:
[tex]\[ 0, \quad 2 - 2(1) = 0, \quad -8 - 2(-2) = -8 + 4 = -4, \quad -8 - 2(-4) = -8 + 8 = 0 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 1 & -2 & | & -4 \\ 0 & 0 & -4 & | & 0 \end{pmatrix} \][/tex]
### Step 3: Forming Pivot in the Third Column
Finally, for the third row, we make the leading coefficient 1.
[tex]\[ R3 \gets -\frac{1}{4} R3 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 1 & -2 & | & -4 \\ 0 & 0 & 1 & | & 0 \end{pmatrix} \][/tex]
### Step 4: Back Substitution for Solution
With the matrix in row echelon form, we use back substitution to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
[tex]\[ \begin{cases} z = 0 \\ y - 2z = -4 \implies y - 0 = -4 \implies y = -4 \\ -x + y + z = -2 \implies -x - 4 + 0 = -2 \implies -x - 4 = -2 \implies -x = 2 \implies x = -2 \end{cases} \][/tex]
So, the solutions to the system of equations are:
[tex]\[ x = -2, \quad y = -4, \quad z = 0 \][/tex]
However, note that during the Gaussian Elimination, we observed a problematic row:
[tex]\[ [nan, nan, nan, nan] \][/tex]
This indicates an inconsistency or an element where division is undefined. If we applied our steps correctly, it implies the system might be inconsistent or degenerate. Hence, this requires a deeper look or alternative methods for confirmation.
[tex]\[ \begin{cases} -x + y + z = -2 \\ -x + 4y - 5z = -14 \\ 4x - 2y - 8z = 0 \end{cases} \][/tex]
We'll start by writing the augmented matrix of the system:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ -1 & 4 & -5 & | & -14 \\ 4 & -2 & -8 & | & 0 \end{pmatrix} \][/tex]
### Step 1: Forming Pivot in the First Column
We'll transform the matrix to get a leading 1 in the first row of the first column. For simplicity, let the first row remain the same, and we'll perform row operations to eliminate the variable [tex]\(x\)[/tex] from the second and third rows.
[tex]\[ R1: \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ -1 & 4 & -5 & | & -14 \\ 4 & -2 & -8 & | & 0 \end{pmatrix} \][/tex]
[tex]\[ R2 \gets R2 - R1 \][/tex]
Second row operation:
[tex]\[ -1 + 1 = 0, \quad 4 - 1 = 3, \quad -5 - 1 = -6, \quad -14 + 2 = -12 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 3 & -6 & | & -12 \\ 4 & -2 & -8 & | & 0 \end{pmatrix} \][/tex]
[tex]\[ R3 \gets R3 + 4 \cdot R1 \][/tex]
Third row operation:
[tex]\[ 4 + 4(-1) = 0, \quad -2 + 4(1) = 2, \quad -8 + 4(1) = 2, \quad 0 + 4(-2) = -8 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 3 & -6 & | & -12 \\ 0 & 2 & -8 & | & -8 \end{pmatrix} \][/tex]
### Step 2: Forming Pivot in the Second Column
Next, we'll make sure the second row has a leading one in the second column.
[tex]\[ R2 \gets \frac{1}{3} R2 \][/tex]
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 1 & -2 & | & -4 \\ 0 & 2 & -8 & | & -8 \end{pmatrix} \][/tex]
To eliminate the variable [tex]\(y\)[/tex] from the third row, we perform:
[tex]\[ R3 \gets R3 - 2 \times R2 \][/tex]
Third row operation:
[tex]\[ 0, \quad 2 - 2(1) = 0, \quad -8 - 2(-2) = -8 + 4 = -4, \quad -8 - 2(-4) = -8 + 8 = 0 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 1 & -2 & | & -4 \\ 0 & 0 & -4 & | & 0 \end{pmatrix} \][/tex]
### Step 3: Forming Pivot in the Third Column
Finally, for the third row, we make the leading coefficient 1.
[tex]\[ R3 \gets -\frac{1}{4} R3 \][/tex]
Resulting in:
[tex]\[ \begin{pmatrix} -1 & 1 & 1 & | & -2 \\ 0 & 1 & -2 & | & -4 \\ 0 & 0 & 1 & | & 0 \end{pmatrix} \][/tex]
### Step 4: Back Substitution for Solution
With the matrix in row echelon form, we use back substitution to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
[tex]\[ \begin{cases} z = 0 \\ y - 2z = -4 \implies y - 0 = -4 \implies y = -4 \\ -x + y + z = -2 \implies -x - 4 + 0 = -2 \implies -x - 4 = -2 \implies -x = 2 \implies x = -2 \end{cases} \][/tex]
So, the solutions to the system of equations are:
[tex]\[ x = -2, \quad y = -4, \quad z = 0 \][/tex]
However, note that during the Gaussian Elimination, we observed a problematic row:
[tex]\[ [nan, nan, nan, nan] \][/tex]
This indicates an inconsistency or an element where division is undefined. If we applied our steps correctly, it implies the system might be inconsistent or degenerate. Hence, this requires a deeper look or alternative methods for confirmation.
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