Final answer:
The determinant of the coefficient matrix for the system of equations is 130, calculated using the standard determinant formula for a 3x3 matrix.
Step-by-step explanation:
To find the determinant of the coefficient matrix of the given system of linear equations, we first need to write down the coefficient matrix and then calculate its determinant. The system given is:
- 4x + 3y + 2z = 0
- -3x + y + 5z = 0
- -x - 4y + 3z = 0
The coefficient matrix A for this system is:
| 4 3 2 |
| -3 1 5 |
| -1 -4 3 |
To find the determinant of matrix A, we can use the following formula:
det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)
Where aij represents the element in the ith row and jth column of matrix A.
Plugging in the values from our matrix, we get:
det(A) = 4((1)(3) - (5)(-4)) - 3((-3)(3) - (5)(-1)) + 2((-3)(-4) - (1)(-1))
det(A) = 4(3 + 20) - 3(-9 + 5) + 2(12 + 1)
det(A) = 4(23) - 3(-4) + 2(13)
det(A) = 92 + 12 + 26
det(A) = 130