use inverse method to solve the system of linear equations
x+y+z-4 =0
2x-3y+4y-33=0
3x -2y -2z -2= 0
To solve the system of linear equations using the inverse method, we need to first write the system in matrix form.
The given system of equations can be written as:
Equation 1: x + y + z = 4 (1)
Equation 2: 2x - 3y + 4z = 33 (2)
Equation 3: 3x - 2y - 2z = 2 (3)
Now, let's define the coefficient matrix A and the variable matrix X as follows:
A = [1 1 1; 2 -3 4; 3 -2 -2]
X = [x; y; z]
We can write the system of equations in matrix form as AX = B, where B is an unknown matrix.
B = [4; 33; 2]
To find the inverse of matrix A, we need to follow these steps:
Step 1: Find the determinant of matrix A. If the determinant is zero, then the inverse does not exist.
Step 2: If the determinant is non-zero, find the adjoint of matrix A.
Step 3: Multiply the adjoint by the reciprocal of the determinant.
Let's calculate the inverse of matrix A using these steps:
Step 1: Determinant of A
det(A) = (1*(-3)*(-2)) + (1*(4)*3) + (1*2*(-2))
= 6 + 12 - 4
= 14
Since the determinant is non-zero (14≠0), we can proceed to the next step.
Step 2: Adjoint of A
The adjoint of matrix A is obtained by swapping the elements of the main diagonal and changing the sign of the other elements.
Adjoint(A) = [(-3)(-2) (-2)(-2) (4)(-2); (4)(3) (3)(-2) (1)(-2); (1)(2) (2)(1) (1)(-3)]
= [6 4 -8; 12 -6 -2; 2 2 -3]
Step 3: Inverse of A
A^(-1) = (1/det(A)) * Adjoint(A)
= (1/14) * [6 4 -8; 12 -6 -2; 2 2 -3]
= 1/14 * [6 4 -8; 12 -6 -2; 2 2 -3]
Now, we can find the solution matrix X by multiplying the inverse of A with B:
X = A^(-1) * B
= [1/14 * [6 4 -8; 12 -6 -2; 2 2 -3]] * [4; 33; 2]
To calculate this multiplication, perform the matrix multiplication of the inverse of A with B.