Solving the System of Equations using Matrix Method:

To solve the given system of equations using the matrix method, we will represent the system in matrix form and then use matrix operations to find the values of x, y, and z.

Step 1: Write the system of equations in matrix form:

The given system of equations can be written in matrix form as:

| 1 1 1 | | x | | 6 |
| 0 1 3 | x | y | = |11 |
| 1 -2 1 | | z | | 0 |

Here, the coefficients of x, y, and z are arranged in a matrix on the left-hand side, and the constants are arranged in a column matrix on the right-hand side.

Step 2: Set up the augmented matrix:

To solve the system using matrix operations, we set up the augmented matrix by combining the coefficient matrix and the constant matrix:

| 1 1 1 | 6 |
| 0 1 3 | 11 |
| 1 -2 1 | 0 |

Step 3: Perform row operations to get the matrix in row-echelon form:

We can use row operations to transform the augmented matrix into row-echelon form. The goal is to create zeros below the pivot elements (the leading non-zero elements in each row) as we move downward.

Performing row operations, we get:

| 1 1 1 | 6 |
| 0 1 3 | 11 |
| 0 -3 0 | -6 |

After performing the row operations, the matrix is in row-echelon form.

Step 4: Perform back substitution to solve for the variables:

Starting from the bottom row, we can solve for z:

-3z = -6
z = 2

Substituting the value of z back into the second row, we can solve for y:

y + 3(2) = 11
y + 6 = 11
y = 5

Finally, substituting the values of y and z into the first row, we can solve for x:

x + 5 + 2 = 6
x + 7 = 6
x = -1

Step 5: Solution:

The solution to the system of equations is x = -1, y = 5, and z = 2.

Summary:

To solve the given system of equations using the matrix method, we represented the system in matrix form and set up the augmented matrix. By performing row operations, we transformed the matrix into row-echelon form. Then, we performed back substitution to find the values of x, y, and z. The final solution to the system of equations is x = -1, y = 5, and z = 2.