Math Problem Statement
Determine the rank of following matrices by reducing them to echlon form
1 2 -1 3
A= 3 -1 2 1
2 -2 3 2
1 -1 1 -1
Solution
We can determine the rank of matrix by reducing it to row echelon form (REF). The rank is the number of non-zero rows in the matrix after it has been reduced.
The matrix is given as:
1 & 2 & -1 & 3 \\ 3 & -1 & 2 & 1 \\ 2 & -2 & 3 & 2 \\ 1 & -1 & 1 & -1 \end{pmatrix}$$ ### Step 1: Use the first row to eliminate the elements below the first pivot (the element in position (1,1)). Subtract 3 times the first row from the second row, 2 times the first row from the third row, and 1 times the first row from the fourth row. $$R_2 = R_2 - 3R_1, \quad R_3 = R_3 - 2R_1, \quad R_4 = R_4 - R_1$$ This gives: $$\begin{pmatrix} 1 & 2 & -1 & 3 \\ 0 & -7 & 5 & -8 \\ 0 & -6 & 5 & -4 \\ 0 & -3 & 2 & -4 \end{pmatrix}$$ ### Step 2: Use the second row to eliminate the elements below the second pivot (the element in position (2,2)). First, subtract $$\frac{6}{7}$$ times the second row from the third row, and $$\frac{3}{7}$$ times the second row from the fourth row. $$R_3 = R_3 - \frac{6}{7} R_2, \quad R_4 = R_4 - \frac{3}{7} R_2$$ This gives: $$\begin{pmatrix} 1 & 2 & -1 & 3 \\ 0 & -7 & 5 & -8 \\ 0 & 0 & \frac{1}{7} & \frac{10}{7} \\ 0 & 0 & \frac{1}{7} & \frac{2}{7} \end{pmatrix}$$ ### Step 3: Use the third row to eliminate the elements below the third pivot (the element in position (3,3)). Subtract the third row from the fourth row. $$R_4 = R_4 - R_3$$ This gives: $$\begin{pmatrix} 1 & 2 & -1 & 3 \\ 0 & -7 & 5 & -8 \\ 0 & 0 & \frac{1}{7} & \frac{10}{7} \\ 0 & 0 & 0 & -\frac{8}{7} \end{pmatrix}$$ ### Step 4: Simplify the pivots. Multiply the third row by 7 and the fourth row by $$-\frac{7}{8}$$ to make the pivots 1: $$\begin{pmatrix} 1 & 2 & -1 & 3 \\ 0 & -7 & 5 & -8 \\ 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ ### Step 5: The rank of the matrix The matrix now has 4 non-zero rows, indicating that all the rows are independent. **The rank of the matrix is 4.** --- Would you like more details, or do you have any questions? Here are five follow-up questions you may find useful: 1. What is the difference between row echelon form (REF) and reduced row echelon form (RREF)? 2. How do elementary row operations affect the determinant of a matrix? 3. Can a matrix have full rank and still be singular? 4. How is the rank of a matrix related to its null space? 5. How can you determine the rank of a matrix using determinants? **Tip:** The rank of a matrix is crucial in understanding the solution space of a system of linear equations—especially when determining whether a system has unique, infinite, or no solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Formulas
Row reduction to echelon form
Pivot position
Elementary row operations
Theorems
Rank theorem (Dimension Theorem)
Gauss-Jordan elimination
Suitable Grade Level
Undergraduate Level (Linear Algebra)