Final answer:
To solve the given system of equations using the inverse of the coefficient matrix, you need to convert the system into matrix form AX = B, compute the inverse of the coefficient matrix A, and then multiply the inverse with the constants matrix B to get the solution matrix X.
Step-by-step explanation:
To solve the system of equations using the inverse of the coefficient matrix, you will first need to express the system in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of the variables, and B is the column matrix of constants. For the given system:
- x + 2y + 3z = 12
- 2x + 3y + 4z = -9
- -x - 2y - 4z = -3
The matrix form would be:
A = [[1, 2, 3],[2, 3, 4],[-1, -2, -4]],
X = [[x], [y], [z]],
B = [[12], [-9], [-3]]
Next, you will calculate the inverse of matrix A, denoted as A⁻¹, using a mathematical software or a calculator. Once you have A⁻¹, you can find the solution by multiplying it with matrix B to get :
X = A⁻¹B
The resulting matrix X will provide you with the values of x, y, and z that solve the system.