Math Problem Statement
Find the value of a and b such that the rank of the matrix A = [[1, -2, 3, 1], [2, 1, -1, 2], [6, -2, a, b]] is 2.
Solution
To solve this problem, we need to determine the values of and such that the rank of the given matrix is 2. Here's the process:
Matrix :
1 & -2 & 3 & 1 \\ 2 & 1 & -1 & 2 \\ 6 & -2 & a & b \end{bmatrix}$$ #### Step 1: Definition of Rank The rank of a matrix is the maximum number of linearly independent rows or columns. If the rank is 2, then at least one of the rows (or columns) must be linearly dependent on the others. This means the determinant of any 3x3 submatrix containing all rows must be zero. #### Step 2: Perform Row Reduction We'll use row reduction to simplify the matrix and analyze the conditions on $$a$$ and $$b$$ for the rank to be 2. 1. Start with the original matrix: $$\begin{bmatrix} 1 & -2 & 3 & 1 \\ 2 & 1 & -1 & 2 \\ 6 & -2 & a & b \end{bmatrix}$$ 2. Eliminate the first column below the pivot (Row 1): - Replace $$R_2$$ with $$R_2 - 2R_1$$. - Replace $$R_3$$ with $$R_3 - 6R_1$$. The resulting matrix is: $$\begin{bmatrix} 1 & -2 & 3 & 1 \\ 0 & 5 & -7 & 0 \\ 0 & 10 & a - 18 & b - 6 \end{bmatrix}$$ 3. Eliminate the second column below the pivot in $$R_2$$: - Replace $$R_3$$ with $$R_3 - 2R_2$$. The resulting matrix is: $$\begin{bmatrix} 1 & -2 & 3 & 1 \\ 0 & 5 & -7 & 0 \\ 0 & 0 & (a - 18) + 14 & b - 6 \end{bmatrix}$$ #### Step 3: Condition for Rank 2 For the matrix to have rank 2: - The third row must be linearly dependent, i.e., all elements in the third row must be zero. This means: $$(a - 18) + 14 = 0 \quad \text{and} \quad b - 6 = 0.$$ #### Step 4: Solve the Equations 1. From $$(a - 18) + 14 = 0$$: $$a - 4 = 0 \implies a = 4.$$ 2. From $$b - 6 = 0$$: $$b = 6.$$ ### Final Answer: $$a = 4, \quad b = 6.$$ Let me know if you need further clarification or a detailed explanation of any step! Here are five follow-up questions: 1. What is the rank of a matrix, and how is it determined? 2. How does row reduction help in determining the rank of a matrix? 3. Why does the determinant of a submatrix relate to the rank of the entire matrix? 4. How can this approach extend to higher-dimensional matrices? 5. What is the geometric interpretation of rank in terms of vector spaces? **Tip**: Always check the determinant of smaller submatrices when verifying rank conditions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Rank
Row Reduction
Determinants
Formulas
Row reduction steps to achieve echelon form
Determinant condition for linear dependence
Theorems
Rank of a matrix theorem
Conditions for linear dependence
Suitable Grade Level
Undergraduate or Advanced High School