Solve the following system of equation, using inverse method if it exists.
x1 + 4x2 + 3x3 = 12
-x1 - 2x2 = -12
2x1 + 2x2 + 3x3 = 8
To solve the system of equations using the inverse method, we will first write the system of equations in matrix form. Let A be the matrix of coefficients, X be the column vector of variables, and B be the column vector of constants.
The given system of equations can be written as:
A * X = B
Where,
A = [[1, 4, 3], [-1, -2, 0], [2, 2, 3]]
X = [x1, x2, x3]
B = [12, -12, 8]
To solve for X, we need to find the inverse of matrix A. If the inverse exists, we can find X using the equation:
X = A^(-1) * B
To find the inverse of A, we will first calculate the determinant of A.
det(A) = 1 * (-2*3 - 2*0) - 4 * (-1*3 - 2*2) + 3 * (-1*2 - (-2)*2)
= 1 * (-6) - 4 * (-7) + 3 * (-2)
= -6 + 28 - 6
= 16
Since the determinant is non-zero (det(A) ≠ 0), the inverse of A exists.
Next, we will calculate the cofactor matrix of A. The cofactor of each element in A is calculated as the determinant of the submatrix obtained by removing the row and column containing that element.
Cofactor of A = [[-2, -20, 4], [-4, -9, 2], [-2, -4, 2]]
To find the adjugate of A, we will transpose the cofactor matrix.
Adjugate of A = [[-2, -4, -2], [-20, -9, -4], [4, 2, 2]]
Finally, we can find the inverse of A by multiplying the adjugate by the reciprocal of the determinant:
A^(-1) = (1/16) * [[-2, -4, -2], [-20, -9, -4], [4, 2, 2]]
= [[-1/8, -1/4, -1/8], [-5/8, -9/16, -1/4], [1/4, 1/8, 1/8]]
Now, we can find X by multiplying the inverse of A by B:
X = A^(-1) * B
= [[-1/8, -1/4, -1/8], [-5/8, -9/16, -1/4], [1/4, 1/8, 1/8]] * [12, -12, 8]
Calculating the matrix multiplication:
X = [((-1/8) * 12) + ((-1/4) * (-12)) + ((-1/8) * 8), ((-5/8) * 12) + ((-9/16) * (-12)) + ((-1/4) * 8), ((1/4) * 12) + ((1/8) * (-12)) + ((1/8) * 8)]
Simplifying:
X = [(-3/2) + (3/4) + (-1), (-15/4) + (9/8) + (-2), (3) + (-3/2) + (1)]
Calculating:
X = [-2/4, (-30/8), (7/2)]
Simplifying:
X = [-1/2, -15/4, 7/2]
Therefore, the solution to the system of equations is:
x1 = -1/2
x2 = -15/4
x3 = 7/2