Math Problem Statement
- 𝐴 = [1 2 3 2 1 4 3 0 5 ]. Find the Rank of matrix
Solution
The given matrix is:
1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{bmatrix}$$ To find the rank of the matrix, we follow these steps: 1. **Form the augmented matrix** and perform row-reducing operations (Gaussian elimination) to get the matrix into its row echelon form (REF). $$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{bmatrix}$$ We'll perform the following operations: - Subtract 2 times the first row from the second row. - Subtract 3 times the first row from the third row. After these operations: $$A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -2 \\ 0 & -6 & -4 \end{bmatrix}$$ 2. Next, we'll simplify the third row by subtracting 2 times the second row from the third row. $$A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -2 \\ 0 & 0 & 0 \end{bmatrix}$$ 3. Now, the matrix is in row echelon form (REF). From this, we can see that there are two non-zero rows. Therefore, the rank of the matrix is **2**. ### Final Answer: The rank of the matrix $$A$$ is **2**. --- Would you like a detailed breakdown of the steps? Here are five related questions for practice: 1. How do you determine the rank of a matrix using Gaussian elimination? 2. What is the difference between row echelon form and reduced row echelon form? 3. Can a matrix have a rank higher than the number of its rows? 4. How do elementary row operations affect the rank of a matrix? 5. What is the relationship between the rank of a matrix and its determinant? **Tip:** The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Gaussian Elimination
Formulas
Rank = number of non-zero rows in row echelon form
Theorems
Rank Theorem
Gaussian Elimination
Suitable Grade Level
Undergraduate Mathematics