Math Problem Statement
Solution
The given problem involves solving the matrix equation , where matrices and are provided. Here's the step-by-step solution:
Step 1: Write down the matrices
Matrix and are:
B = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}.$$ ### Step 2: Transpose matrix $$A$$ The transpose of $$A$$, denoted $$A^T$$, is: $$A^T = \begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}.$$ ### Step 3: Find the inverse of matrix $$B$$ The inverse of $$B$$, denoted $$B^{-1}$$, is calculated as: $$B^{-1} = \frac{1}{\det(B)} \cdot \text{adj}(B),$$ where: - $$\det(B) = (1)(2) - (1)(1) = 2 - 1 = 1$$, - $$\text{adj}(B) = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}$$. Thus: $$B^{-1} = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}.$$ ### Step 4: Solve for $$X$$ The equation $$A^T X = B^{-1}$$ can be rewritten as: $$X = (A^T)^{-1} B^{-1}.$$ #### (i) Find $$(A^T)^{-1}$$: The inverse of $$A^T$$ is calculated similarly: $$A^T = \begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}.$$ - $$\det(A^T) = (2)(0) - (1)(-1) = 0 + 1 = 1$$, - $$\text{adj}(A^T) = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}$$. Thus: $$(A^T)^{-1} = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}.$$ #### (ii) Compute $$X = (A^T)^{-1} B^{-1}$$: Substitute the values of $$(A^T)^{-1}$$ and $$B^{-1}$$: $$X = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}.$$ Perform matrix multiplication: $$X = \begin{pmatrix} (0)(2) + (1)(-1) & (0)(-1) + (1)(1) \\ (-1)(2) + (2)(-1) & (-1)(-1) + (2)(1) \end{pmatrix} = \begin{pmatrix} -1 & 1 \\ -4 & 3 \end{pmatrix}.$$ ### Final Answer: $$X = \begin{pmatrix} -1 & 1 \\ -4 & 3 \end{pmatrix}.$$ --- Would you like me to: 1. Verify the solution through substitution? 2. Explain any step in more detail? 3. Discuss how matrix operations work? 4. Extend the explanation to other matrix equations? 5. Solve a different example? ### Tip: When solving matrix equations, always check determinants to ensure the inverses exist!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Transposition
Matrix Inverse
Matrix Multiplication
Formulas
Matrix Transpose: (A^T)_{ij} = A_{ji}
Matrix Inverse: A^{-1} = (1/det(A)) * adj(A)
Matrix Multiplication: (AB)_{ij} = Σ A_{ik} B_{kj}
Theorems
Matrix Inverse Theorem
Determinant and Adjoint Relation
Suitable Grade Level
Undergraduate (Linear Algebra)
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