Respuesta :
Answer:
The system has infinitely many solutions
[tex]\begin{array}{ccc}x_1&=&-x_3\\x_2&=&-x_3\\x_3&=&arbitrary\end{array}[/tex]
Step-by-step explanation:
Gauss–Jordan elimination is a method of solving a linear system of equations. This is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.
An Augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
There are three elementary matrix row operations:
- Switch any two rows
- Multiply a row by a nonzero constant
- Add one row to another
To solve the following system
[tex]\begin{array}{ccccc}x_1&-3x_2&-2x_3&=&0\\-x_1&2x_2&x_3&=&0\\2x_1&+3x_2&+5x_3&=&0\end{array}[/tex]
Step 1: Transform the augmented matrix to the reduced row echelon form
[tex]\left[ \begin{array}{cccc} 1 & -3 & -2 & 0 \\\\ -1 & 2 & 1 & 0 \\\\ 2 & 3 & 5 & 0 \end{array} \right][/tex]
This matrix can be transformed by a sequence of elementary row operations
Row Operation 1: add 1 times the 1st row to the 2nd row
Row Operation 2: add -2 times the 1st row to the 3rd row
Row Operation 3: multiply the 2nd row by -1
Row Operation 4: add -9 times the 2nd row to the 3rd row
Row Operation 5: add 3 times the 2nd row to the 1st row
to the matrix
[tex]\left[ \begin{array}{cccc} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \end{array} \right][/tex]
The reduced row echelon form of the augmented matrix is
[tex]\left[ \begin{array}{cccc} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \end{array} \right][/tex]
which corresponds to the system
[tex]\begin{array}{ccccc}x_1&&-x_3&=&0\\&x_2&+x_3&=&0\\&&0&=&0\end{array}[/tex]
The system has infinitely many solutions.
[tex]\begin{array}{ccc}x_1&=&-x_3\\x_2&=&-x_3\\x_3&=&arbitrary\end{array}[/tex]
