Solving a System of Linear Equations using Matrices and Gaussian Elimination

Math Problem Statement

x_1[2,2,3]+x_2[1,3,-2]+x_3[3,5,1]=[-16,-28,-3] Solve:

We have a system of three equations with three unknowns:

[ x_1 [2,2,3] + x_2 [1,3,-2] + x_3 [3,5,1] = [-16, -28, -3] ]

Writing each equation separately:

$2x_1 + x_2 + 3x_3 = -16$ $2x_1 + 3x_2 + 5x_3 = -28$ $3x_1 - 2x_2 + x_3 = -3$

We can solve this system using the matrix method.

2 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & -2 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -16 \\ -28 \\ -3 \end{bmatrix}$$ #### Solve using Gaussian elimination or inverse matrix: Let's compute the solution. numpy as np # Define the coefficient matrix A and the constant matrix B A = np.array([[2, 1, 3], [2, 3, 5], [3, -2, 1]]) B = np.array([-16, -28, -3]) # Solve the system Ax = B solution = np.linalg.solve(A, B) solutionThe solution to the system is: $$x_1 = -3.5, \quad x_2 = -4.5, \quad x_3 = -1.5$$ Would you like a step-by-step explanation of how this was solved? ### Related Questions: 1. How can Gaussian elimination be used to solve this system? 2. What is the determinant of the coefficient matrix, and what does it tell us? 3. How can this system be solved using Cramer's rule? 4. What happens if we modify one of the equations slightly? 5. How does matrix inversion work in solving such systems? **Tip:** If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many solutions!

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Linear Algebra
Systems of Equations
Matrix Operations
Gaussian Elimination

Ax = B
Gaussian Elimination
Inverse Matrix Method

Cramer's Rule
Matrix Inversion Theorem

Grades 10-12

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