Math Problem Statement
3 -1 2 -6 2 4 3 1 2 Find rank of matrix
Solution
To find the rank of a matrix, we need to reduce it to its row echelon form (or reduced row echelon form) and count the number of non-zero rows. Let's work through the matrix:
3 & -1 & 2 \\ -6 & 2 & 4 \\ 3 & 1 & 2 \end{pmatrix}$$ ### Step 1: Apply Row Operations We start by eliminating the entries below the first pivot, which is 3 in the first row. 1. **Row 2:** To eliminate the first element of the second row, add 2 times the first row to the second row: $$R_2 \to R_2 + 2R_1$$ $$\begin{pmatrix} 3 & -1 & 2 \\ 0 & 0 & 8 \\ 3 & 1 & 2 \end{pmatrix}$$ 2. **Row 3:** To eliminate the first element of the third row, subtract the first row from the third row: $$R_3 \to R_3 - R_1$$ $$\begin{pmatrix} 3 & -1 & 2 \\ 0 & 0 & 8 \\ 0 & 2 & 0 \end{pmatrix}$$ ### Step 2: Further Simplify Now, we focus on the third row. The second pivot is 2 in the third row. To simplify further: 1. **Row 3:** Divide the third row by 2 to get a simpler form: $$R_3 \to \frac{1}{2}R_3$$ $$\begin{pmatrix} 3 & -1 & 2 \\ 0 & 0 & 8 \\ 0 & 1 & 0 \end{pmatrix}$$ 2. **Row 2:** Eliminate the last element in row 2 by subtracting 8 times row 3 from row 2: $$R_2 \to R_2 - 8R_3$$ $$\begin{pmatrix} 3 & -1 & 2 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$$ ### Step 3: Count Non-Zero Rows The matrix is now in row echelon form. The non-zero rows are: $$\begin{pmatrix} 3 & -1 & 2 \\ 0 & 1 & 0 \end{pmatrix}$$ There are **2** non-zero rows, so the rank of the matrix is **2**. --- Do you need further details or have any questions? Here are 5 related questions you can explore: 1. What is the significance of a matrix's rank? 2. How is the rank of a matrix connected to its determinant? 3. What happens when the rank of a matrix equals the number of columns? 4. How can the nullity of a matrix be found using its rank? 5. Can the rank of a matrix be greater than the number of rows? **Tip:** Always ensure that the matrix is simplified correctly to row echelon form when calculating its rank.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Matrix Operations
Formulas
Row Echelon Form
Pivot Elements
Theorems
Rank Theorem
Row Reduction
Suitable Grade Level
Grades 11-12, Undergraduate