Math Problem Statement
solve this:
Find the rank and the nullity of the given matrix:
A = [[0, 0, 1, -3, -6], [0, 1, 3, -6, -9], [1, 0, 0, -3, -5]]
Solution
To solve this problem, we will determine the rank and nullity of the matrix :
Matrix :
\begin{bmatrix} 0 & 0 & 1 & -3 & -6 \\ 0 & 1 & 3 & -6 & -9 \\ 1 & 0 & 0 & -3 & -5 \end{bmatrix}$$ #### Step 1: Put the matrix into row-echelon form We perform row operations to simplify the matrix into row-echelon form. 1. **Initial Matrix**: $$A = \begin{bmatrix} 0 & 0 & 1 & -3 & -6 \\ 0 & 1 & 3 & -6 & -9 \\ 1 & 0 & 0 & -3 & -5 \end{bmatrix}$$ 2. Swap Row 1 and Row 3 to ensure the first pivot is non-zero: $$A = \begin{bmatrix} 1 & 0 & 0 & -3 & -5 \\ 0 & 1 & 3 & -6 & -9 \\ 0 & 0 & 1 & -3 & -6 \end{bmatrix}$$ 3. Eliminate the $$a_{31}$$ term (already 0), so no changes are needed. #### Step 2: Verify pivots The resulting matrix has pivots in columns 1, 2, and 3. Hence: - The **rank** of $$A$$ is 3. #### Step 3: Compute nullity The nullity of a matrix is given by: $$\text{Nullity}(A) = n - \text{Rank}(A)$$ where $$n$$ is the number of columns. Here, $$n = 5$$ and $$\text{Rank}(A) = 3$$. Thus: $$\text{Nullity}(A) = 5 - 3 = 2$$ ### Final Answer: - Rank: $$\text{Rank}(A) = 3$$ - Nullity: $$\text{Nullity}(A) = 2$$ Would you like a detailed explanation of how row operations were performed? Let me know! --- ### Related Questions: 1. What are the steps to reduce a matrix to row-echelon form? 2. How does the rank-nullity theorem work in linear algebra? 3. How can you verify the solution by finding the null space of the matrix? 4. What is the geometrical interpretation of rank and nullity? 5. How does the number of rows affect the rank of a matrix? ### Tip: To quickly check the nullity, remember that the null space of a matrix is related to the number of free variables in its solution set.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Row Echelon Form
Rank and Nullity
Formulas
Nullity(A) = n - Rank(A)
Elementary row operations for matrix reduction
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate (First-Year University Level)