Math Problem Statement
Given the equation P = 2Q^T, find the value of c, where P and Q are matrices.
Solution
The problem shown in the image involves two matrices, and , and a relationship between them. Let's break it down:
We are given the equation:
Where:
\quad \text{and} \quad Q = \begin{pmatrix} 2c - 3b & 2a + 1 \\ a & b + 7 \end{pmatrix}$$ Our task is to find the value of $$c$$. ### Steps: 1. **Transpose of Matrix $$Q$$:** $$Q^T = \begin{pmatrix} 2c - 3b & a \\ 2a + 1 & b + 7 \end{pmatrix}$$ 2. **Multiplying by 2 (since $$P = 2Q^T$$):** $$2Q^T = 2 \times \begin{pmatrix} 2c - 3b & a \\ 2a + 1 & b + 7 \end{pmatrix} = \begin{pmatrix} 4c - 6b & 2a \\ 4a + 2 & 2b + 14 \end{pmatrix}$$ 3. **Equating matrices $$P$$ and $$2Q^T$$:** $$\begin{pmatrix} a & q \\ 2b & 3c \end{pmatrix} = \begin{pmatrix} 4c - 6b & 2a \\ 4a + 2 & 2b + 14 \end{pmatrix}$$ Now, we can compare the elements of the matrices: - $$a = 4c - 6b$$ - $$q = 2a$$ - $$2b = 4a + 2$$ - $$3c = 2b + 14$$ We can solve this system of equations to find $$c$$. I will now proceed with solving it.The solution to the system of equations is: - $$a = 2$$ - $$b = 5$$ - $$c = 8$$ Thus, the value of $$c$$ is $$8$$. Would you like more details on how this was solved, or do you have any other questions? Here are 5 related questions: 1. How do you find the transpose of a matrix? 2. Can we solve a system of equations without a matrix? 3. What is the general process of matrix multiplication? 4. What are the properties of a transposed matrix? 5. How would the solution change if the equation were different, say $$P = 3Q^T$$? **Tip:** Always check for matrix dimensions before performing matrix operations to ensure the operations are valid.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Transposition
Matrix Multiplication
Linear Equations
Formulas
Matrix Transposition: (A^T)_{ij} = A_{ji}
Matrix Multiplication
Solving systems of linear equations
Theorems
Matrix Transposition Properties
Matrix Multiplication Properties
Suitable Grade Level
Grades 10-12