Solve the following equation using Cramer’s rule:

2x+y-z=3; x+y+z=1; x-2y-3z=4

### Solution to the System of Equations Using Cramer’s Rule

The given system of equations is:

1. \( 2x + y - z = 3 \)
2. \( x + y + z = 1 \)
3. \( x - 2y - 3z = 4 \)

Using Cramer's Rule, the solution to this system of linear equations involves finding the determinants of matrices. The general form of the equations can be expressed as \( AX = B \), where:

\[
A = \begin{bmatrix} 
2 & 1 & -1 \\ 
1 & 1 & 1 \\ 
1 & -2 & -3 
\end{bmatrix}, \quad 
X = \begin{bmatrix} 
x \\ 
y \\ 
z 
\end{bmatrix}, \quad 
B = \begin{bmatrix} 
3 \\ 
1 \\ 
4 
\end{bmatrix}.
\]

#### Step 1: Compute the Determinant of \( A \)

The determinant of \( A \) is calculated as follows:

\[
\text{Det}(A) = 
\begin{vmatrix} 
2 & 1 & -1 \\ 
1 & 1 & 1 \\ 
1 & -2 & -3 
\end{vmatrix}.
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