Math Problem Statement
Consider the following matrices.
A = 1 −2 −4 6 , B = 1 −2 0 −2
Find an elementary matrix E such that EA = B
Solution
To solve for the elementary matrix such that , we need to determine what row operations were applied to matrix to transform it into matrix . Elementary matrices correspond to these row operations.
Matrices A and B:
1 & -2 \\ -4 & 6 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -2 \\ 0 & -2 \end{pmatrix}$$ #### Step 1: Identify row operations We can see that matrix $$B$$ is obtained by applying the following row operation to $$A$$: - **Row 2** in matrix $$B$$ is a result of adding 4 times row 1 to row 2 in matrix $$A$$. So the row operation is: $$R_2 \rightarrow R_2 + 4R_1$$ #### Step 2: Express the row operation as an elementary matrix The elementary matrix corresponding to the row operation $$R_2 \rightarrow R_2 + 4R_1$$ is: $$E = \begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix}$$ #### Step 3: Verify $$E \times A = B$$ Now, let's verify that multiplying $$E$$ with $$A$$ results in $$B$$. $$EA = \begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} 1 & -2 \\ -4 & 6 \end{pmatrix} = \begin{pmatrix} (1)(1) + (0)(-4) & (1)(-2) + (0)(6) \\ (4)(1) + (1)(-4) & (4)(-2) + (1)(6) \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ 0 & -2 \end{pmatrix}$$ This is exactly matrix $$B$$. Thus, the elementary matrix $$E$$ is: $$E = \begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix}$$ Would you like further details on elementary matrices or any other clarifications? Here are 5 related questions to expand on this topic: 1. How do elementary matrices relate to row operations? 2. Can every matrix be transformed into another by an elementary matrix? 3. What are the different types of elementary row operations? 4. How would you compute the inverse of an elementary matrix? 5. What happens when multiple elementary matrices are applied successively? **Tip**: When multiplying matrices, always ensure that the dimensions are compatible: the number of columns in the first matrix must match the number of rows in the second matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Elementary Matrices
Matrix Row Operations
Formulas
Elementary matrix corresponding to row operation: E = R_i → R_i + kR_j
Matrix multiplication formula: (EA = B)
Theorems
Elementary Matrices and Row Operations Theorem
Suitable Grade Level
Undergraduate (Linear Algebra Course)
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