Math Problem Statement
Solve
If ρ(A) = 2, then K has the value in the matrix A as shown.
Solution
We are given a matrix and asked to find the value of such that the rank of matrix , denoted , is 2.
The matrix is:
-1 & 2 & 1 & -2 \\ 2 & -4 & -2 & K \\ 0 & 1 & -3 & 2 \end{pmatrix}$$ The rank of a matrix is the number of linearly independent rows or columns in the matrix. Since $$\rho(A) = 2$$, this means that there should be two linearly independent rows in the matrix, and the third row must be a linear combination of the other two rows. ### Step-by-step solution: 1. **Row 1**: $$(-1, 2, 1, -2)$$ 2. **Row 2**: $$(2, -4, -2, K)$$ 3. **Row 3**: $$(0, 1, -3, 2)$$ We will perform elementary row operations to simplify the matrix and check for conditions that make the rank equal to 2. ### Step 1: Row Reduction We will start by making Row 2 a multiple of Row 1. Multiply Row 1 by 2: $$\text{Row 1} \times 2 = (-2, 4, 2, -4)$$ Now add this to Row 2: $$\text{Row 2} + \text{Row 1} \times 2 = (2, -4, -2, K) + (-2, 4, 2, -4) = (0, 0, 0, K - 4)$$ Thus, after this operation, the matrix becomes: $$A = \begin{pmatrix} -1 & 2 & 1 & -2 \\ 0 & 0 & 0 & K - 4 \\ 0 & 1 & -3 & 2 \end{pmatrix}$$ ### Step 2: Analyze the Rank Condition - For the rank to be 2, the second row $$(0, 0, 0, K-4)$$ should be a zero row, because this would mean that Row 3 must be dependent on the remaining non-zero rows. Therefore, $$K - 4 = 0$$, which gives: $$K = 4$$ Thus, the correct value of $$K$$ is $$4$$, which corresponds to option **D**. Would you like a detailed breakdown of the steps, or do you have any other questions? Here are 5 related questions to consider: 1. What is the definition of matrix rank and how is it determined? 2. How do elementary row operations affect the rank of a matrix? 3. What happens when two rows of a matrix are linearly dependent? 4. How can matrix rank help in solving systems of linear equations? 5. How does the concept of rank apply to invertibility of matrices? **Tip:** When solving matrix rank problems, row reduction (Gaussian elimination) is a very useful technique to simplify the analysis.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Rank
Elementary Row Operations
Linear Dependence
Formulas
Rank of matrix
Row operations for Gaussian elimination
Theorems
Rank Theorem
Suitable Grade Level
College/University Level