Final answer:
To write the system as a matrix equation, the coefficient matrix is [3 2 1; -2 -1 4; 1 -1 32] and the answer matrix is [-8; 7; 15]. The determinant of the coefficient matrix is 100.
Step-by-step explanation:
To write the system of equations as a matrix equation, we can set up a coefficient matrix and an answer matrix. The coefficient matrix contains the coefficients of the variables (x, y, z), and the answer matrix contains the constants on the right side of the equations (-8, 7, 15). The matrix equation for this system is:
[3 2 1; -2 -1 4; 1 -1 32] [x; y; z] = [-8; 7; 15]
To find the determinant of the coefficient matrix, we can use the formula for a 3x3 matrix determinant:
det([3 2 1; -2 -1 4; 1 -1 32]) = (3)(-1)(32) + (2)(4)(1) + (1)(-2)(-1) - (1)(-1)(32) - (2)(-1)(1) - (3)(-1)(1) = 100