To solve the system of equations:

$w - 2y + z = 0$ $-x - 3y + z = 3$ $-2w - x + y - z = 3$ $3x + 9y = -12$

We can represent this system of equations as a matrix equation:

$\mathbf{A} \cdot \mathbf{x} = \mathbf{b}$

Where:

• $\mathbf{A}$ is the coefficient matrix,
• $\mathbf{x}$ is the column vector of variables $[w, x, y, z]^T$,
• $\mathbf{b}$ is the column vector of constants on the right-hand side of the equations.

Step 1: Write the augmented matrix

We begin by writing the system as an augmented matrix:

1 & 0 & -2 & 1 & | & 0 \\ -1 & -3 & 0 & 1 & | & 3 \\ -2 & -1 & 1 & -1 & | & 3 \\ 0 & 3 & 9 & 0 & | & -12 \end{bmatrix}$$ ### Step 2: Perform row operations (Gaussian Elimination) We apply row operations to reduce the augmented matrix into row echelon form or reduced row echelon form. This process involves adding or subtracting rows to eliminate variables and solve for the unknowns. ### Step 3: Solve the system Once the matrix is in reduced form, we can extract the solution. #### Matrix Computation: Let me work through the row reduction process now. sympy as sp # Define variables w, x, y, z = sp.symbols('w x y z') # Augmented matrix from the system of equations A = sp.Matrix([ [1, 0, -2, 1, 0], [-1, -3, 0, 1, 3], [-2, -1, 1, -1, 3], [0, 3, 9, 0, -12] ]) # Perform row reduction rref_matrix, pivot_columns = A.rref() rref_matrixThe reduced row echelon form (RREF) of the augmented matrix is: $$\begin{bmatrix} 1 & 0 & 0 & 0 & | & -1 \\ 0 & 1 & 0 & 0 & | & -1 \\ 0 & 0 & 1 & 0 & | & -1 \\ 0 & 0 & 0 & 1 & | & -1 \end{bmatrix}$$ This indicates the following solution: $$w = -1, \quad x = -1, \quad y = -1, \quad z = -1$$ Thus, the solution to the system of equations is the vector $$\mathbf{x} = [-1, -1, -1, -1]^T$$. If you'd like, I can help you with a more detailed explanation of this row reduction process or any other queries you have. Let me know! ### 5 Follow-up Questions: 1. What is the significance of the reduced row echelon form in solving linear systems? 2. Can you explain how row operations work in Gaussian elimination? 3. How would you solve a system with no solution using this method? 4. What would the solution look like if the system had infinitely many solutions? 5. How can we represent solutions to systems of equations using vector notation? **Tip:** In linear algebra, row reduction can also help determine the rank of the matrix, which tells us about the number of independent equations in the system!