Math Problem Statement
Diketahui persamaan matriks A, B, dan C: A = [[2, -1], [1, 4]], B = [[x - y, 2], [3, v]], C = [[7, 2], [3, 1]]. Jika B - A = C^T dan C merupakan transpose matriks C, maka nilai x, y sama dengan?
Solution
The problem states that we are given matrices , , and as follows:
B = \begin{bmatrix} x - y & 2 \\ 3 & v \end{bmatrix}, \quad C = \begin{bmatrix} 7 & 2 \\ 3 & 1 \end{bmatrix}$$ We are also given that: $$B - A = C^T$$ where $$C^T$$ is the transpose of matrix $$C$$. ### Step-by-step solution: 1. **Transpose of matrix $$C$$:** $$C^T = \begin{bmatrix} 7 & 3 \\ 2 & 1 \end{bmatrix}$$ 2. **Equation $$B - A = C^T$$:** $$\begin{bmatrix} x - y & 2 \\ 3 & v \end{bmatrix} - \begin{bmatrix} 2 & -1 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 7 & 3 \\ 2 & 1 \end{bmatrix}$$ This gives us the system of equations by subtracting matrix $$A$$ from $$B$$: $$\begin{bmatrix} (x - y) - 2 & 2 - (-1) \\ 3 - 1 & v - 4 \end{bmatrix} = \begin{bmatrix} 7 & 3 \\ 2 & 1 \end{bmatrix}$$ 3. **Solving for each element:** - For the top left element: $$(x - y) - 2 = 7$$ $$x - y = 9$$ - For the top right element: $$2 - (-1) = 3$$ (this is consistent). - For the bottom left element: $$3 - 1 = 2$$ (this is also consistent). - For the bottom right element: $$v - 4 = 1$$ $$v = 5$$ 4. **Conclusion:** We now have the equation $$x - y = 9$$ and the value $$v = 5$$. Therefore, the value of $$x$$ and $$y$$ is such that: $$x - y = 9$$ Would you like to explore the solution further or need any additional clarification? --- Here are some follow-up questions based on this problem: 1. How do we solve for $$x$$ and $$y$$ given more information? 2. What are the properties of matrix transposition used in this problem? 3. How do matrix subtraction and addition work element-wise? 4. Can you generalize this process for matrices of larger dimensions? 5. How does the matrix transpose change the orientation of elements? **Tip**: Always verify your matrix operations by checking each element individually when solving matrix equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Matrix Transposition
Algebraic Equations
Formulas
Matrix transposition: C^T
Matrix subtraction: B - A
Solving algebraic systems
Theorems
Properties of matrix transposition
Matrix equality theorem
Suitable Grade Level
Grades 10-12