Math Problem Statement
Find the rank and nullity of the matrix
Find the rank and nullity of the matrix
Solution
To find the rank and nullity of the given matrix , we need to perform row reduction (Gaussian elimination) to bring the matrix into its row echelon form (or reduced row echelon form). From there, the rank is determined by the number of non-zero rows, and the nullity is given by the difference between the number of columns and the rank.
The matrix is:
1 & 2 & 1 & 0 \\ 3 & 2 & 1 & 2 \\ 2 & -1 & 2 & 5 \\ 5 & 6 & 3 & 2 \\ 1 & 3 & -1 & -3 \end{pmatrix}$$ ### Steps to solve: 1. **Gaussian elimination**: - Use row operations to reduce the matrix to its echelon form. 2. **Determine rank**: - The rank is the number of non-zero rows in the row echelon form of the matrix. 3. **Nullity**: - Nullity = (Number of columns) - (Rank). Let me compute the row echelon form to determine the rank and nullity.The **rank** of the given matrix $$A$$ is 3, and the **nullity** is 1. This means there are 3 linearly independent rows in the matrix, and the solution space to $$A \mathbf{x} = 0$$ has dimension 1. Would you like more details on any step, or do you have any specific questions about the process? Here are five related questions to further your understanding: 1. What is the significance of the rank in terms of linear independence? 2. How does the nullity relate to the number of solutions of a linear system? 3. What does the row echelon form of a matrix represent? 4. How can the nullity help determine if a system has free variables? 5. How would changing a row or column affect the rank of a matrix? **Tip:** The sum of the rank and the nullity of a matrix equals the number of its columns, according to the rank-nullity theorem.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Nullity
Gaussian Elimination
Formulas
Nullity = Number of columns - Rank
Rank-Nullity Theorem
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate Level