Math Problem Statement
- Evaluate the Rank of the matrix by reducing it to the Normal form(2,3,1,−1)(−1,−1,−2,−43 )(16 ,33,−20,−7)
Solution
To determine the rank of the matrix, we can reduce it to its row echelon form (or normal form). The rank of a matrix is defined as the number of linearly independent rows (or columns), which corresponds to the number of non-zero rows in the row echelon form.
Let's start by writing down the matrix:
2 & 3 & 1 & -1 \\ -1 & -1 & -2 & -43 \\ 16 & 33 & -20 & -7 \end{bmatrix}$$ We’ll perform row operations to reduce this matrix to row echelon form. ### Step-by-Step Solution #### Step 1: Make the first element of the first row 1 Divide the first row by 2: $$\begin{bmatrix} 1 & 1.5 & 0.5 & -0.5 \\ -1 & -1 & -2 & -43 \\ 16 & 33 & -20 & -7 \end{bmatrix}$$ #### Step 2: Make the elements below the leading 1 in column 1 equal to 0 Add the first row to the second row: $$\begin{bmatrix} 1 & 1.5 & 0.5 & -0.5 \\ 0 & 0.5 & -1.5 & -43.5 \\ 16 & 33 & -20 & -7 \end{bmatrix}$$ Subtract 16 times the first row from the third row: $$\begin{bmatrix} 1 & 1.5 & 0.5 & -0.5 \\ 0 & 0.5 & -1.5 & -43.5 \\ 0 & 9 & -28 & 1 \end{bmatrix}$$ #### Step 3: Make the second pivot element equal to 1 Divide the second row by 0.5: $$\begin{bmatrix} 1 & 1.5 & 0.5 & -0.5 \\ 0 & 1 & -3 & -87 \\ 0 & 9 & -28 & 1 \end{bmatrix}$$ #### Step 4: Make elements above and below the pivot in column 2 equal to 0 Subtract 1.5 times the second row from the first row: $$\begin{bmatrix} 1 & 0 & 5 & 130 \\ 0 & 1 & -3 & -87 \\ 0 & 9 & -28 & 1 \end{bmatrix}$$ Subtract 9 times the second row from the third row: $$\begin{bmatrix} 1 & 0 & 5 & 130 \\ 0 & 1 & -3 & -87 \\ 0 & 0 & -1 & 784 \end{bmatrix}$$ #### Step 5: Make the third pivot element equal to 1 Multiply the third row by -1: $$\begin{bmatrix} 1 & 0 & 5 & 130 \\ 0 & 1 & -3 & -87 \\ 0 & 0 & 1 & -784 \end{bmatrix}$$ #### Step 6: Make elements above the pivot in column 3 equal to 0 Add 3 times the third row to the second row: $$\begin{bmatrix} 1 & 0 & 5 & 130 \\ 0 & 1 & 0 & -2339 \\ 0 & 0 & 1 & -784 \end{bmatrix}$$ Subtract 5 times the third row from the first row: $$\begin{bmatrix} 1 & 0 & 0 & 4050 \\ 0 & 1 & 0 & -2339 \\ 0 & 0 & 1 & -784 \end{bmatrix}$$ ### Conclusion The matrix is now in row echelon form, and each row has a leading 1 in a unique column. Since there are three non-zero rows, the rank of the matrix is **3**. Let me know if you'd like further details on this solution or have any questions! --- ### Relative Questions 1. How do you find the rank of a matrix using column operations? 2. What is the rank of a zero matrix? 3. Can a matrix have a rank greater than its smallest dimension? 4. How does the determinant relate to the rank of a matrix? 5. What are the practical applications of finding a matrix's rank? --- **Tip:** The rank of a matrix can also be determined by counting the non-zero rows in its reduced row echelon form (RREF).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Echelon Form
Formulas
Row Echelon Form
Rank of a Matrix (number of linearly independent rows)
Theorems
Rank-Nullity Theorem
Suitable Grade Level
Grades 11-12 or College Level