Math Problem Statement
reduce the matrix into normal form and find its rank
Reduce the matrix into normal form and find its rank.
Solution
To reduce the given matrix to its row echelon form (REF) and find its rank, let's first extract the matrix from the image.
The matrix is:
1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 5 & 5 \\ 3 & -4 & 5 & 8 \end{bmatrix}$$ ### Step 1: Perform row operations to reduce the matrix to row echelon form. I'll walk through each operation. #### Original matrix: $$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 5 & 5 \\ 3 & -4 & 5 & 8 \end{bmatrix}$$ #### Row Operations: 1. Subtract row 1 from row 2: $$R_2 = R_2 - R_1 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 2 & 3 & 5 & 5 \\ 3 & -4 & 5 & 8 \end{bmatrix}$$ 2. Subtract 2 times row 1 from row 3: $$R_3 = R_3 - 2R_1 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 3 \\ 3 & -4 & 5 & 8 \end{bmatrix}$$ 3. Subtract 3 times row 1 from row 4: $$R_4 = R_4 - 3R_1 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 3 \\ 0 & -7 & 2 & 5 \end{bmatrix}$$ 4. Subtract row 2 from row 3: $$R_3 = R_3 - R_2 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & -7 & 2 & 5 \end{bmatrix}$$ 5. Add 7 times row 2 to row 4: $$R_4 = R_4 + 7R_2 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 16 & 26 \end{bmatrix}$$ 6. Subtract 16 times row 3 from row 4: $$R_4 = R_4 - 16R_3 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 26 \end{bmatrix}$$ 7. Divide row 4 by 26 to normalize: $$R_4 = \frac{1}{26} R_4 \implies \begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ ### Row Echelon Form: The matrix in row echelon form (REF) is: $$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ ### Rank of the Matrix: The rank of a matrix is the number of non-zero rows in its row echelon form. In this case, all 4 rows are non-zero. Hence, the rank of the matrix is **4**. Would you like any further details or have any questions? --- Here are five questions related to this process: 1. What is the difference between row echelon form and reduced row echelon form? 2. How does the rank of a matrix affect the solutions of a system of linear equations? 3. Can a matrix have a rank greater than its number of rows or columns? 4. How do you find the determinant of a matrix, and how does rank relate to it? 5. What are some practical applications of matrix rank in data analysis? **Tip**: To confirm the rank of a matrix, you can also use the determinant of the largest non-zero submatrix (if applicable).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Row Echelon Form
Rank of a Matrix
Formulas
Elementary Row Operations
Rank = Number of Non-Zero Rows in Row Echelon Form
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Grades 11-12, Undergraduate