To find the determinant of the coefficient matrix of the given system of equations, follow these steps:
The system of equations is:
[tex]\[
\left\{
\begin{array}{l}
4x + 3y + 2z = 0 \\
-3x + y + 5z = 0 \\
-x - 4y + 3z = 0
\end{array}
\right.
\][/tex]
The coefficient matrix [tex]\( A \)[/tex] is:
[tex]\[
A =
\begin{pmatrix}
4 & 3 & 2 \\
-3 & 1 & 5 \\
-1 & -4 & 3
\end{pmatrix}
\][/tex]
To find the determinant of matrix [tex]\( A \)[/tex], we use the following formula for the determinant of a 3x3 matrix:
[tex]\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
where the matrix [tex]\( A \)[/tex] is:
[tex]\[
A =
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\][/tex]
For our matrix [tex]\( A \)[/tex], we have:
[tex]\[
a = 4, \quad b = 3, \quad c = 2 \\
d = -3, \quad e = 1, \quad f = 5 \\
g = -1, \quad h = -4, \quad i = 3
\][/tex]
Plugging these values into the determinant formula, we get:
[tex]\[
\text{det}(A) = 4(1 \cdot 3 - 5 \cdot -4) - 3(-3 \cdot 3 - 5 \cdot -1) + 2(-3 \cdot -4 - 1 \cdot -1)
\][/tex]
Simplify each term:
[tex]\[
1 \cdot 3 - 5 \cdot -4 = 3 + 20 = 23 \\
-3 \cdot 3 - 5 \cdot -1 = -9 + 5 = -4 \\
-3 \cdot -4 - 1 \cdot -1 = 12 + 1 = 13
\][/tex]
Therefore:
[tex]\[
\text{det}(A) = 4 \cdot 23 - 3 \cdot -4 + 2 \cdot 13
\][/tex]
[tex]\[
= 92 + 12 + 26
\][/tex]
[tex]\[
= 130
\][/tex]
Thus, the determinant of the coefficient matrix is:
[tex]\[
\boxed{130}
\][/tex]
