Math Problem Statement
what is the rank 1 1 −3 −1 8 0 7 0 0
Solution
The given matrix is:
1 & 1 & -3 \\ -1 & 8 & 0 \\ 7 & 0 & 0 \end{bmatrix}$$ To find the **rank** of the matrix, we need to determine the number of **linearly independent rows** or, equivalently, the number of **non-zero rows** in the row echelon form (REF) or reduced row echelon form (RREF). Let's perform row reduction to convert the matrix into echelon form: ### Step 1: Matrix $$\begin{bmatrix} 1 & 1 & -3 \\ -1 & 8 & 0 \\ 7 & 0 & 0 \end{bmatrix}$$ ### Step 2: Eliminate the entries below the first pivot (which is 1). - Add row 1 to row 2. - Subtract 7 times row 1 from row 3. $$\begin{bmatrix} 1 & 1 & -3 \\ 0 & 9 & -3 \\ 0 & -7 & 21 \end{bmatrix}$$ ### Step 3: Eliminate the entry below the second pivot (which is 9). - Add $$\frac{7}{9} \times \text{row 2}$$ to row 3. $$\begin{bmatrix} 1 & 1 & -3 \\ 0 & 9 & -3 \\ 0 & 0 & 19 \end{bmatrix}$$ ### Step 4: Simplify pivots by dividing rows 2 and 3 by their leading entries. $$\begin{bmatrix} 1 & 1 & -3 \\ 0 & 1 & -\frac{1}{3} \\ 0 & 0 & 1 \end{bmatrix}$$ ### Conclusion: The matrix is now in echelon form, and we observe three non-zero rows. Therefore, the matrix has **3 pivots** (1 in each row). ### Final Answer: The **rank** of the matrix is **3**. --- Would you like further details on any of the steps, or have any questions? Here are some related questions to expand on the topic: 1. What does it mean for a matrix to have a certain rank? 2. How do you determine if rows are linearly independent? 3. How does the rank relate to the solutions of a system of equations? 4. What is the significance of row echelon form in linear algebra? 5. How does rank affect the invertibility of a matrix? **Tip:** The rank of a matrix is equal to the number of linearly independent rows or columns, whichever is fewer.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Linear Independence
Formulas
Row Reduction
Rank = number of linearly independent rows
Theorems
Rank-Nullity Theorem
Echelon Form Theorem
Suitable Grade Level
Grades 11-12 or College Level