Math Problem Statement
-1 - 3 3 - 1
1 1 - 1 0
2 - 5 2 - 3
-1 1 0 1
Reduce the matrix into echelon form and find the rank of matrix
Solution
To reduce the given matrix into echelon form and determine its rank, let's follow these steps:
Given Matrix
-1 & -3 & 3 & -1 \\ 1 & 1 & -1 & 0 \\ 2 & -5 & 2 & -3 \\ -1 & 1 & 0 & 1 \end{pmatrix}$$ ### Step 1: Row Operations to Get Echelon Form #### 1. First, let's make the first element of the first row (pivot) positive by swapping Row 1 with Row 2. $$\text{Swap } R_1 \text{ and } R_2: \begin{pmatrix} 1 & 1 & -1 & 0 \\ -1 & -3 & 3 & -1 \\ 2 & -5 & 2 & -3 \\ -1 & 1 & 0 & 1 \end{pmatrix}$$ #### 2. Next, add multiples of the first row to the others to eliminate the first column elements below the pivot. - $$R_2 \rightarrow R_2 + R_1$$ - $$R_3 \rightarrow R_3 - 2R_1$$ - $$R_4 \rightarrow R_4 + R_1$$ $$\begin{pmatrix} 1 & 1 & -1 & 0 \\ 0 & -2 & 2 & -1 \\ 0 & -7 & 4 & -3 \\ 0 & 2 & -1 & 1 \end{pmatrix}$$ #### 3. Now, proceed with the second pivot (second element in the second row). Divide Row 2 by $$-2$$ to make the pivot element $$1$$. $$R_2 \rightarrow -\frac{1}{2} R_2: \begin{pmatrix} 1 & 1 & -1 & 0 \\ 0 & 1 & -1 & \frac{1}{2} \\ 0 & -7 & 4 & -3 \\ 0 & 2 & -1 & 1 \end{pmatrix}$$ #### 4. Eliminate the elements below and above the second pivot (second column). - $$R_3 \rightarrow R_3 + 7R_2$$ - $$R_4 \rightarrow R_4 - 2R_2$$ - $$R_1 \rightarrow R_1 - R_2$$ $$\begin{pmatrix} 1 & 0 & 0 & -\frac{1}{2} \\ 0 & 1 & -1 & \frac{1}{2} \\ 0 & 0 & -3 & \frac{1}{2} \\ 0 & 0 & 1 & 0 \end{pmatrix}$$ #### 5. Make the third pivot $$1$$ by dividing Row 3 by $$-3$$. $$R_3 \rightarrow -\frac{1}{3} R_3: \begin{pmatrix} 1 & 0 & 0 & -\frac{1}{2} \\ 0 & 1 & -1 & \frac{1}{2} \\ 0 & 0 & 1 & -\frac{1}{6} \\ 0 & 0 & 1 & 0 \end{pmatrix}$$ #### 6. Finally, eliminate the element above the third pivot in Row 2. $$R_2 \rightarrow R_2 + R_3: \begin{pmatrix} 1 & 0 & 0 & -\frac{1}{2} \\ 0 & 1 & 0 & \frac{1}{3} \\ 0 & 0 & 1 & -\frac{1}{6} \\ 0 & 0 & 0 & \frac{1}{6} \end{pmatrix}$$ ### Step 2: Identify Echelon Form and Rank The matrix is now in echelon form. The rank of the matrix is the number of non-zero rows. #### Rank of Matrix: **4** (since all rows are non-zero). Would you like more details on any of the steps? Here are some related questions you might find interesting: 1. What are the types of row operations in Gaussian elimination? 2. How do you interpret the rank of a matrix in terms of linear independence? 3. What is the significance of a matrix's rank in determining the solutions of a linear system? 4. How can you determine if a matrix is singular using its echelon form? 5. How does Gaussian elimination help in finding the inverse of a matrix? **Tip:** The rank of a matrix is crucial in determining whether a system of linear equations has a unique solution, infinitely many solutions, or no solution at all.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Operations
Gaussian Elimination
Echelon Form
Rank of a Matrix
Formulas
-
Theorems
-
Suitable Grade Level
Advanced High School