Math Problem Statement
Find the rank of the matrix [ 1 2 3] [1 4 2] [2 6 5]
Solution
To find the rank of a matrix, we perform row reduction (Gaussian elimination) to convert it to its row echelon form (REF). The rank is the number of non-zero rows in the row echelon form.
The given matrix is:
1 & 2 & 3 \\ 1 & 4 & 2 \\ 2 & 6 & 5 \end{bmatrix}$$ ### Step 1: Subtract the first row from the second row $$R_2 = R_2 - R_1$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & -1 \\ 2 & 6 & 5 \end{bmatrix}$$ ### Step 2: Subtract 2 times the first row from the third row $$R_3 = R_3 - 2R_1$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & -1 \\ 0 & 2 & -1 \end{bmatrix}$$ ### Step 3: Subtract the second row from the third row $$R_3 = R_3 - R_2$$ $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & -1 \\ 0 & 0 & 0 \end{bmatrix}$$ The matrix is now in row echelon form. There are 2 non-zero rows, so the rank of the matrix is: $$\text{Rank} = 2$$ Would you like further details or have any questions? Here are 5 questions that expand on this topic: 1. What are the differences between row echelon form and reduced row echelon form (RREF)? 2. How is the rank related to the solutions of a system of linear equations? 3. Can the rank of a matrix be greater than its number of rows or columns? 4. What is the connection between the determinant of a matrix and its rank? 5. How do you determine the nullity of a matrix using its rank? **Tip:** The rank of a matrix is always equal to the number of leading entries (pivots) in its row echelon form.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Gaussian Elimination
Row Echelon Form (REF)
Formulas
Row operations for Gaussian elimination
Theorems
Rank Theorem (The rank of a matrix is the number of leading entries in its row echelon form)
Suitable Grade Level
Grades 10-12 or College Level