Math Problem Statement
Use row operations to change the matrix below to reduced form: [ 1 2 -2 -1 ; 0 3 -15 1 ; 0 -1 5 -1/3 ].
Solution
The problem asks you to perform row operations to reduce the matrix to its row-reduced form. Let's go step by step and solve this.
The matrix given is:
1 & 2 & -2 & -1 \\ 0 & 3 & -15 & 1 \\ 0 & -1 & 5 & -\frac{1}{3} \end{pmatrix}$$ ### Step 1: The goal is to get zeros below and above the pivots and reduce the matrix to its simplest form. - The pivot in the first column is already $$1$$ in row 1, so we don’t need to change row 1. ### Step 2: Eliminate the non-zero entry in column 2 below the pivot. - To eliminate the $$-1$$ in row 3, column 2, we can add $$\frac{1}{3}$$ times row 2 to row 3: $$R_3 = R_3 + \frac{1}{3}R_2$$ Performing this gives the updated matrix: $$\begin{pmatrix} 1 & 2 & -2 & -1 \\ 0 & 3 & -15 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ This is the reduced row echelon form (RREF). Would you like further details on any specific part of the process? Here are 5 follow-up questions: 1. What are the different types of elementary row operations? 2. How is the row-reduced form of a matrix used to solve systems of linear equations? 3. What are the conditions for a matrix to be in row-reduced form? 4. Can every matrix be reduced to row-reduced form? 5. How does the number of pivots in the reduced matrix relate to the solution of a system? **Tip:** Always aim to use the smallest row operations to simplify computations when performing Gaussian elimination.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Row Operations
Row-Reduced Echelon Form
Formulas
Row Operations
Matrix Reduction
Theorems
Row-Reduced Echelon Form (RREF)
Gaussian Elimination
Suitable Grade Level
Grades 10-12, College