Math Problem Statement
Find the rank of the matrix A = [ [-2, -1, -3, -1], [1, 2, 3, 1], [0, 1, 0, 1] ] by reducing it to row echelon form.
Solution
Let's go through the problems in this image, beginning with the first one.
Problem 1: Rank of the Matrix
We are given the matrix:
-2 & -1 & -3 & -1 \\ 1 & 2 & 3 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix}$$ We need to reduce this matrix to its row echelon form to find the rank. Let me work on this reduction. ### Step 1: Original Matrix $$A = \begin{pmatrix} -2 & -1 & -3 & -1 \\ 1 & 2 & 3 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix}$$ ### Step 2: Row Operations to Reduce to Row Echelon Form We will perform elementary row operations to transform the matrix into upper triangular form. 1. Multiply Row 1 by $$-\frac{1}{2}$$: $$R_1 \to \frac{1}{2}R_1: \begin{pmatrix} 1 & \frac{1}{2} & \frac{3}{2} & \frac{1}{2} \end{pmatrix}$$ 2. Subtract Row 1 from Row 2 to eliminate the first element in the second row: $$R_2 \to R_2 - R_1: \begin{pmatrix} 0 & \frac{3}{2} & \frac{3}{2} & \frac{1}{2} \end{pmatrix}$$ 3. The third row already has a 0 in the first column, so no further operations are needed for that. ### Step 3: Continue Reduction Next, we'll make the second row's leading coefficient 1 by multiplying by $$\frac{2}{3}$$: $$R_2 \to \frac{2}{3}R_2: \begin{pmatrix} 0 & 1 & 1 & \frac{1}{3} \end{pmatrix}$$ Finally, subtract $$\frac{1}{2}$$ of Row 2 from Row 1 to eliminate the second column of Row 1: $$R_1 \to R_1 - \frac{1}{2}R_2: \begin{pmatrix} 1 & 0 & 1 & \frac{1}{3} \end{pmatrix}$$ Thus, the row-echelon form is: $$\text{Row-echelon form of } A = \begin{pmatrix} 1 & 0 & 1 & \frac{1}{3} \\ 0 & 1 & 1 & \frac{1}{3} \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ ### Step 4: Determine Rank The rank is the number of non-zero rows in this row-echelon form. Since there are two non-zero rows, the **rank of matrix A is 2**. --- Would you like details for the next problems, or do you have any questions on this solution? Here are 5 questions to expand on this topic: 1. What does the rank of a matrix represent in terms of its columns? 2. Can a matrix have a rank higher than its number of rows or columns? 3. How is the rank of a matrix related to the solution of a system of linear equations? 4. What is the difference between row echelon form and reduced row echelon form? 5. How does the rank relate to the determinant of a matrix? **Tip**: The rank of a matrix gives you insight into the number of linearly independent rows or columns, which is key for determining the solution space of associated linear systems.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Formulas
Row operations for Gaussian elimination
Theorems
Rank-Nullity Theorem
Suitable Grade Level
University level (Linear Algebra)
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