Answer:
The system has infinitely many solutions
![\begin{array}{ccc}x_1&=&-x_3\\x_2&=&-x_3\\x_3&=&arbitrary\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccc%7Dx_1%26%3D%26-x_3%5C%5Cx_2%26%3D%26-x_3%5C%5Cx_3%26%3D%26arbitrary%5Cend%7Barray%7D)
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
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccc%7Dx_1%26-3x_2%26-2x_3%26%3D%260%5C%5C-x_1%262x_2%26x_3%26%3D%260%5C%5C2x_1%26%2B3x_2%26%2B5x_3%26%3D%260%5Cend%7Barray%7D)
Step 1: Transform the augmented matrix to the reduced row echelon form
![\left[ \begin{array}{cccc} 1 & -3 & -2 & 0 \\\\ -1 & 2 & 1 & 0 \\\\ 2 & 3 & 5 & 0 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcccc%7D%201%20%26%20-3%20%26%20-2%20%26%200%20%5C%5C%5C%5C%20-1%20%26%202%20%26%201%20%26%200%20%5C%5C%5C%5C%202%20%26%203%20%26%205%20%26%200%20%5Cend%7Barray%7D%20%5Cright%5D)
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
![\left[ \begin{array}{cccc} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcccc%7D%201%20%26%200%20%26%201%20%26%200%20%5C%5C%5C%5C%200%20%26%201%20%26%201%20%26%200%20%5C%5C%5C%5C%200%20%26%200%20%26%200%20%26%200%20%5Cend%7Barray%7D%20%5Cright%5D)
The reduced row echelon form of the augmented matrix is
![\left[ \begin{array}{cccc} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcccc%7D%201%20%26%200%20%26%201%20%26%200%20%5C%5C%5C%5C%200%20%26%201%20%26%201%20%26%200%20%5C%5C%5C%5C%200%20%26%200%20%26%200%20%26%200%20%5Cend%7Barray%7D%20%5Cright%5D)
which corresponds to the system
![\begin{array}{ccccc}x_1&&-x_3&=&0\\&x_2&+x_3&=&0\\&&0&=&0\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccccc%7Dx_1%26%26-x_3%26%3D%260%5C%5C%26x_2%26%2Bx_3%26%3D%260%5C%5C%26%260%26%3D%260%5Cend%7Barray%7D)
The system has infinitely many solutions.
![\begin{array}{ccc}x_1&=&-x_3\\x_2&=&-x_3\\x_3&=&arbitrary\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccc%7Dx_1%26%3D%26-x_3%5C%5Cx_2%26%3D%26-x_3%5C%5Cx_3%26%3D%26arbitrary%5Cend%7Barray%7D)