Math Problem Statement
Find rank of the matrix A=[1 2 -1 3] [4 1 2 1][3 -1 1 2][1 2 0 1]
Solution
To find the rank of the matrix , we need to determine the maximum number of linearly independent rows or columns in the matrix. Let's go through the steps:
Given matrix:
1 & 2 & -1 & 3 \\ 4 & 1 & 2 & 1 \\ 3 & -1 & 1 & 2 \\ 1 & 2 & 0 & 1 \end{bmatrix}$$ 1. **Form the augmented matrix:** We'll perform row reduction (Gaussian elimination) to transform this matrix into its row echelon form (REF). 2. **Row reduction steps:** - Start with the first element in the first row (pivot = 1): $$R_2 \rightarrow R_2 - 4R_1, \quad R_3 \rightarrow R_3 - 3R_1, \quad R_4 \rightarrow R_4 - R_1$$ After these operations: $$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & -7 & 6 & -11 \\ 0 & -7 & 4 & -7 \\ 0 & 0 & 1 & -2 \end{bmatrix}$$ - Now, take the second row and normalize the pivot (second element of the second row = -7): $$R_2 \rightarrow \frac{R_2}{-7}$$ After normalization: $$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -\frac{6}{7} & \frac{11}{7} \\ 0 & -7 & 4 & -7 \\ 0 & 0 & 1 & -2 \end{bmatrix}$$ - Make the elements below the pivot in column 2 zero: $$R_3 \rightarrow R_3 + 7R_2$$ After this operation: $$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -\frac{6}{7} & \frac{11}{7} \\ 0 & 0 & -\frac{2}{7} & \frac{6}{7} \\ 0 & 0 & 1 & -2 \end{bmatrix}$$ - Now, work on the third row: $$R_3 \rightarrow \frac{R_3}{-\frac{2}{7}}$$ After normalization: $$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -\frac{6}{7} & \frac{11}{7} \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 1 & -2 \end{bmatrix}$$ - Finally, make the element below the third pivot (row 4) zero: $$R_4 \rightarrow R_4 - R_3$$ After this operation: $$\begin{bmatrix} 1 & 2 & -1 & 3 \\ 0 & 1 & -\frac{6}{7} & \frac{11}{7} \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ 3. **Count the non-zero rows:** In the row echelon form, all rows are non-zero, indicating four linearly independent rows. Thus, the rank of matrix $$A$$ is **4**. Would you like more details or have any questions? Here are some related questions to expand your understanding: 1. How does one determine the linear independence of rows or columns in a matrix? 2. What is the significance of a matrix's rank in linear algebra? 3. How would you find the rank of a non-square matrix? 4. What role does the determinant play in finding the rank of a square matrix? 5. How does Gaussian elimination relate to solving systems of linear equations? **Tip:** The rank of a matrix provides insight into the dimension of the vector space spanned by its rows or columns, and it's a crucial concept in understanding solutions to linear systems.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Reduction
Gaussian Elimination
Formulas
Row Echelon Form
Rank = Number of linearly independent rows or columns
Theorems
Rank-Nullity Theorem
Properties of Row Echelon Form
Suitable Grade Level
Grades 10-12 (High School, Early College)