https://file.aiquickdraw.com/mathbot/file/1726756872705h241pofu.jpg

The problem shown in the image involves two matrices, \( P \) and \( Q \), and a relationship between them. Let's break it down:

We are given the equation:

\[
P = 2Q^T
\]

Where:

\[
P = \begin{pmatrix} a & q \\ 2b & 3c \end{pmatrix}
\quad \text{and} \quad
Q = \begin{pmatrix} 2c - 3b & 2a + 1 \\ a & b + 7 \end{pmatrix}
\]

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.

Given the equation P = 2Q^T, find the value of c, where P and Q are matrices.

Learn how to solve for the value of c in the equation P = 2Q^T with step-by-step matrix transposition and multiplication. This problem covers key mathematical concepts like matrix operations and solving systems of linear equations. Suitable for Grades 10-12.

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