Math Problem Statement
Find the rank of the following matrices:-
al [[1, 3, 7, 1, 2], [4, 0, 5, 2, 9], [3, 3, 4, 7, 4], [0, 0, 6, 6, - 3]]
Solution
To find the rank of the matrix, we perform Gaussian elimination to reduce it to row echelon form (REF). The rank of a matrix is the number of non-zero rows in its REF.
The given matrix is:
1 & 3 & 7 & 1 & 2 \\ 4 & 0 & 5 & 2 & 9 \\ 3 & 3 & 4 & 7 & 4 \\ 0 & 0 & 6 & 6 & -3 \end{bmatrix}$$ ### Step 1: Perform Row Operations We start with the matrix and use elementary row operations (subtracting multiples of rows) to create zeros below the pivot in each column. 1. $$R_2 = R_2 - 4R_1$$ 2. $$R_3 = R_3 - 3R_1$$ This will update the matrix as follows: $$\begin{bmatrix} 1 & 3 & 7 & 1 & 2 \\ 0 & -12 & -23 & -2 & 1 \\ 0 & -6 & -17 & 4 & -2 \\ 0 & 0 & 6 & 6 & -3 \end{bmatrix}$$ ### Step 2: Create Zero Below Next Pivot (Second Column) We now focus on the second column. To make the zeros below the pivot (second row), perform the following: 3. $$R_3 = R_3 - \frac{1}{2} R_2$$ This gives: $$\begin{bmatrix} 1 & 3 & 7 & 1 & 2 \\ 0 & -12 & -23 & -2 & 1 \\ 0 & 0 & -5.5 & 5 & -2.5 \\ 0 & 0 & 6 & 6 & -3 \end{bmatrix}$$ ### Step 3: Work on Third Column Finally, we make the matrix triangular by eliminating below the pivot in the third column: 4. $$R_4 = R_4 - \frac{6}{-5.5} R_3$$ This simplifies to: $$\begin{bmatrix} 1 & 3 & 7 & 1 & 2 \\ 0 & -12 & -23 & -2 & 1 \\ 0 & 0 & -5.5 & 5 & -2.5 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Step 4: Count Non-Zero Rows In the final row echelon form, there are **3 non-zero rows**, meaning the rank of the matrix is: $$\text{Rank} = 3$$ --- Would you like further details on the row operations or a more in-depth explanation of the Gaussian elimination process? ### Related Questions: 1. How does the rank of a matrix relate to its solutions in linear systems? 2. Can you explain the difference between row echelon form (REF) and reduced row echelon form (RREF)? 3. What is the impact of having a zero row on the rank of a matrix? 4. How does Gaussian elimination help in solving linear systems? 5. Can the rank of a matrix be determined from its determinant? ### Tip: The rank of a matrix is always less than or equal to the smallest dimension (rows or columns).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Gaussian Elimination
Row Echelon Form
Formulas
Row Operations
Rank = Number of Non-zero Rows in Row Echelon Form
Theorems
Rank Theorem (Dimension Theorem)
Gaussian Elimination Theorem
Suitable Grade Level
Grades 10-12