Reduced Row Echelon Form of a 3x4 Augmented Matrix

Math Problem Statement

Find the reduced row echelon form of the following augmented matrix: [[-1, 1, -4, 28], [4, 0, -1, -7], [0, 4, 2, 10]].

Let's solve the given augmented matrix:

-1 & 1 & -4 & 28 \\ 4 & 0 & -1 & -7 \\ 0 & 4 & 2 & 10 \end{array}\right]$$ We will apply row operations to bring this matrix into reduced row echelon form (RREF). The goal is to make the left-hand side a leading 1 in each row, with zeros elsewhere in the column, while preserving the augmented part on the right. I'll work through the steps and provide the RREF.The reduced row echelon form (RREF) of the given augmented matrix is: $$\left[\begin{array}{ccc|c} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & -5 \end{array}\right]$$ This shows that the system has the solution: $$x_1 = -3, \quad x_2 = 5, \quad x_3 = -5$$ Would you like a detailed explanation of how the row reduction was done? Here are some related questions: 1. How do we interpret a system of equations from an augmented matrix? 2. What is the difference between row echelon form and reduced row echelon form? 3. How do you know when a matrix is in reduced row echelon form? 4. Can all matrices be reduced to RREF? 5. What do the pivot columns tell us in solving a system of equations? **Tip:** When solving systems of equations using matrices, check for free variables if the matrix doesn't have a pivot in every column. This indicates the system might have infinitely many solutions.

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