Math Problem Statement
find the rank of the matrix 2 1 3 5, 4 2 1 3, 8 4 7 13, 8 4 -3-1
Solution
To find the rank of the given matrix, we will perform elementary row operations to convert it into its row echelon form (REF) or reduced row echelon form (RREF). The rank of a matrix is the number of non-zero rows in its REF or RREF.
Given matrix:
2 & 1 & 3 & 5 \\ 4 & 2 & 1 & 3 \\ 8 & 4 & 7 & 13 \\ 8 & 4 & -3 & -1 \end{pmatrix}$$ ### Step 1: Convert to Row Echelon Form We will perform row operations to simplify the matrix. #### 1. Row 2: Subtract $$2 \times \text{Row 1}$$ from Row 2: $$R_2 \rightarrow R_2 - 2R_1$$ $$R_2 = (4, 2, 1, 3) - 2(2, 1, 3, 5) = (4 - 4, 2 - 2, 1 - 6, 3 - 10) = (0, 0, -5, -7)$$ #### 2. Row 3: Subtract $$4 \times \text{Row 1}$$ from Row 3: $$R_3 \rightarrow R_3 - 4R_1$$ $$R_3 = (8, 4, 7, 13) - 4(2, 1, 3, 5) = (8 - 8, 4 - 4, 7 - 12, 13 - 20) = (0, 0, -5, -7)$$ #### 3. Row 4: Subtract $$4 \times \text{Row 1}$$ from Row 4: $$R_4 \rightarrow R_4 - 4R_1$$ $$R_4 = (8, 4, -3, -1) - 4(2, 1, 3, 5) = (8 - 8, 4 - 4, -3 - 12, -1 - 20) = (0, 0, -15, -21)$$ At this stage, the matrix looks like: $$A = \begin{pmatrix} 2 & 1 & 3 & 5 \\ 0 & 0 & -5 & -7 \\ 0 & 0 & -5 & -7 \\ 0 & 0 & -15 & -21 \end{pmatrix}$$ #### 4. Row 4: Subtract $$3 \times \text{Row 2}$$ from Row 4: $$R_4 \rightarrow R_4 - 3R_2$$ $$R_4 = (0, 0, -15, -21) - 3(0, 0, -5, -7) = (0, 0, -15 + 15, -21 + 21) = (0, 0, 0, 0)$$ Now, the matrix becomes: $$A = \begin{pmatrix} 2 & 1 & 3 & 5 \\ 0 & 0 & -5 & -7 \\ 0 & 0 & -5 & -7 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ ### Step 2: Identify the Rank The row echelon form of the matrix has two non-zero rows. Therefore, the rank of the matrix is **2**. --- Would you like more details or have any further questions? Here are 5 questions to deepen your understanding: 1. What is the relationship between the rank of a matrix and its determinant? 2. How can row reduction be used to determine the inverse of a matrix? 3. Can a matrix with more rows than columns have full rank? 4. What is the significance of a zero row in determining the rank? 5. How does the rank of a matrix relate to the solution of a system of linear equations? **Tip:** A matrix's rank is always less than or equal to the smallest of its row or column dimensions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Elementary Row Operations
Formulas
Rank of a matrix = Number of non-zero rows in Row Echelon Form (REF) or Reduced Row Echelon Form (RREF)
Theorems
Rank Theorem
Suitable Grade Level
Grades 10-12