To solve the given system of equations using the inverse matrix method, we need to follow these steps:
1. Write the system of equations in matrix form:
[1 2 3] [x] [20]
[5 -2 4] * [y] = [3]
[-4 4 1] [z] [0]
2. Find the inverse of the coefficient matrix [A] by calculating its determinant and the adjoint matrix:
A = [1 2 3]
[5 -2 4]
[-4 4 1]
det(A) = 1(1*(-2*1 - 4*4) - 2*(5*1 - 4*4) + 3*(5*4 - (-2)*(-4))) = 1(8 + 18 + 60) = 86
Adjoint matrix of A:
adj(A) = [6 -16 -14]
[17 1 18]
[18 22 -8]
Inverse of A:
A^(-1) = adj(A) / det(A)
3. Multiply the inverse of the coefficient matrix by the constant matrix to solve for the variables:
[x] [20] [6]
[y] = [3] * [17]
[z] [0] [18]
Therefore, the solution to the system of equations is:
x = 6, y = 17, z = 18
