To solve the given system of linear equations using matrix methods, we first need to represent the system in matrix form.
The system of equations is:
[tex]\[
\left\{\begin{array}{l}
2x + y + 3z = 13 \\
x + 2y = 11 \\
3x + z = 10
\end{array}\right.
\][/tex]
### Step 1: Create the coefficient matrix
The coefficient matrix [tex]\( A \)[/tex] and the right-hand side matrix [tex]\( B \)[/tex] are:
[tex]\[
A = \begin{bmatrix}
2 & 1 & 3 \\
1 & 2 & 0 \\
3 & 0 & 1
\end{bmatrix}, \quad
B = \begin{bmatrix}
13 \\
11 \\
10
\end{bmatrix}
\][/tex]
### Step 2: Calculate the determinant of the coefficient matrix
The determinant of the coefficient matrix [tex]\( A \)[/tex] is:
[tex]\[
\text{Det}(A) = -15.0
\][/tex]
### Step 3: Solve the system of linear equations
By solving the system, we determine the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex]:
[tex]\[
x = 3.0, \quad y = 4.0, \quad z = 1.0000000000000002
\][/tex]
### Conclusion
Therefore, the answers are:
[tex]\[
\text{The determinant of the coefficient matrix is } -15.0
\][/tex]
[tex]\[
\begin{aligned}
x &= 3.0 \\
y &= 4.0 \\
z &= 1.0000000000000002
\end{aligned}
\][/tex]
