Math Problem Statement
solve this using normal form
Find the rank of a matrix by using Normal Form, A = [1, 1, -1, 1; 1, -1, 2, -1; 3, 1, 0, 1].
Solution
To solve this problem, we need to find the rank of the given matrix using normal form (also known as row echelon form or Gaussian elimination).
Given matrix:
1 & 1 & -1 & 1 \\ 1 & -1 & 2 & -1 \\ 3 & 1 & 0 & 1 \end{bmatrix}$$ The steps to reduce the matrix to row echelon form are: 1. **Start with the first row** and use it to make the leading entry (pivot) in other rows 0. We begin by subtracting appropriate multiples of row 1 from rows 2 and 3. - Row 2: $$R_2 = R_2 - R_1$$ - Row 3: $$R_3 = R_3 - 3R_1$$ After this step, we get: $$A = \begin{bmatrix} 1 & 1 & -1 & 1 \\ 0 & -2 & 3 & -2 \\ 0 & -2 & 3 & -2 \end{bmatrix}$$ 2. **Simplify row 3**: Notice that row 3 is identical to row 2, so we can subtract row 2 from row 3 to make row 3 all zeros: $$R_3 = R_3 - R_2$$ The resulting matrix is: $$A = \begin{bmatrix} 1 & 1 & -1 & 1 \\ 0 & -2 & 3 & -2 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ 3. **Divide row 2** by $$-2$$ to normalize the pivot element: $$R_2 = \frac{1}{-2} R_2 = \begin{bmatrix} 0 & 1 & -\frac{3}{2} & 1 \end{bmatrix}$$ The matrix now looks like: $$A = \begin{bmatrix} 1 & 1 & -1 & 1 \\ 0 & 1 & -\frac{3}{2} & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ 4. **Use row 2** to eliminate the second column entry in row 1. Subtract row 2 from row 1: $$R_1 = R_1 - R_2$$ This gives: $$A = \begin{bmatrix} 1 & 0 & \frac{1}{2} & 0 \\ 0 & 1 & -\frac{3}{2} & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Final Result (Row Echelon Form): $$A = \begin{bmatrix} 1 & 0 & \frac{1}{2} & 0 \\ 0 & 1 & -\frac{3}{2} & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ### Rank of the Matrix: The rank of a matrix is the number of non-zero rows in its row echelon form. In this case, there are **two non-zero rows** in the final matrix, so the rank of matrix $$A$$ is: $$\boxed{2}$$ Would you like further explanation on any step or a related topic? ### Related Questions: 1. How do you find the rank of a matrix using the determinant method? 2. Can the rank of a matrix be greater than the number of rows? 3. How does Gaussian elimination differ from the Gauss-Jordan method for solving systems of equations? 4. What is the significance of the rank of a matrix in terms of linear independence? 5. How can you identify if a matrix is singular based on its rank? #### Tip: Always check your row operations carefully when reducing a matrix to avoid calculation errors—especially when dealing with fractions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Row Echelon Form
Gaussian Elimination
Matrix Rank
Formulas
Row Echelon Form
Rank = Number of non-zero rows
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 10-12 or undergraduate level
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