We are given two matrices $P$ and $Q$, and the condition that $P = Q^T$. The matrices are as follows:

$P = \begin{bmatrix} 1 & 2b \\ -a & -3 \end{bmatrix}, \quad Q = \begin{bmatrix} 4 & -4a \\ 5 & 5 \end{bmatrix}$

Step 1: Write the transpose of $Q$

The transpose of $Q$, denoted $Q^T$, is obtained by swapping the rows and columns of $Q$:

$Q^T = \begin{bmatrix} 4 & 5 \\ -4a & 5 \end{bmatrix}$

Step 2: Set up the equation $P = Q^T$

From the problem, we are given that $P = Q^T$, so we can set the elements of the two matrices equal to each other:

$\begin{bmatrix} 1 & 2b \\ -a & -3 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ -4a & 5 \end{bmatrix}$

Step 3: Equate corresponding elements

By comparing the corresponding elements of the matrices, we obtain the following system of equations:

1. $1 = 4$ (top-left element)
2. $2b = 5$ (top-right element)
3. $-a = -4a$ (bottom-left element)
4. $-3 = 5$ (bottom-right element)

Step 4: Analyze the system of equations

From the first and fourth equations:

1. $1 = 4$ is impossible, so there seems to be a typo or mistake in the problem setup. Let's assume it's a misprint and proceed with the other equations for now.

From the second equation:

$2b = 5 \implies b = \frac{5}{2}$

From the third equation:

$-a = -4a \implies a = 0$

Step 5: Compute $3a + 2b$

Now that we know $a = 0$ and $b = \frac{5}{2}$, we can calculate:

$3a + 2b = 3(0) + 2\left(\frac{5}{2}\right) = 0 + 5 = 5$

Final Answer:

The value of $3a + 2b$ is $\boxed{5}$.

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Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. If $P$ and $Q$ were 3x3 matrices, how would you compute $P = Q^T$?
2. Can you find a condition where two matrices are symmetric?
3. What is the general formula for the transpose of a matrix?
4. How do you verify if two matrices are inverses of each other?
5. What are the properties of matrix multiplication?

Tip: When comparing matrices element-wise, always ensure that both matrices have the same dimensions.