Final answer:
a. The system of equations can be written in matrix form Ax=b as A = [[2, -1, 0], [-2, -2, 1], [0, 1, 3]], x = [[x1], [x2], [x3]], and b = [[30], [50], [140]]. b. The determinant of matrix A is 7, indicating that the system of equations has a unique solution. c. By using Gaussian elimination, we find the solution to be x1 = 10, x2 = -10, and x3 = 50. d. The solution can be verified by substituting the values into the original equations.
Step-by-step explanation:
a. To write the system of equations in matrix form Ax=b, we can arrange the coefficients of the variables in a matrix A, the variables themselves in a matrix x, and the constants on the right-hand side of the equations in a matrix b. The matrices A, x, and b can be defined as follows:
A = [[2, -1, 0], [-2, -2, 1], [0, 1, 3]]
x = [[x1], [x2], [x3]]
b = [[30], [50], [140]]
b. To find the determinant of matrix A, we can use the formula for a 3x3 matrix:
det(A) = 2(3) - (-1)(1) + 0(1) = 6 + 1 + 0 = 7
Since the determinant is non-zero (7 ≠ 0), the system of equations has a unique solution.
c. To solve the system of equations, we can use Gaussian elimination:
Step 1: Add 2 times the first equation to the second equation.
Step 2: Subtract the second equation from 3 times the third equation.
Step 3: Solve the resulting system of equations.
The solution is x1 = 10, x2 = -10, and x3 = 50.
d. To verify the solution, we substitute the values of x1, x2, and x3 into the original equations:
Equation 1: 2(10) - (-10) = 30
Equation 2: -2(10) - 2(-10) + (50) = 50
Equation 3: (-10) + 3(50) = 140
These equations are all true, so the solution is correct.