The problem can be interpreted as:

We are given the equation:

$3Q - \begin{pmatrix} 2 & 1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 4 & 11 \\ 14 & 6 \end{pmatrix}$

We need to find matrix $Q$.

Step-by-Step Solution:

1. Rearrange the equation to solve for $Q$:

$3Q = \begin{pmatrix} 4 & 11 \\ 14 & 6 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ -5 & 3 \end{pmatrix}$

2. Perform matrix addition:

$3Q = \begin{pmatrix} 4 + 2 & 11 + 1 \\ 14 + (-5) & 6 + 3 \end{pmatrix}$ $3Q = \begin{pmatrix} 6 & 12 \\ 9 & 9 \end{pmatrix}$

3. Divide the matrix by 3 to solve for $Q$:

$Q = \frac{1}{3} \begin{pmatrix} 6 & 12 \\ 9 & 9 \end{pmatrix}$ $Q = \begin{pmatrix} 2 & 4 \\ 3 & 3 \end{pmatrix}$

Thus, matrix $Q$ is:

$Q = \begin{pmatrix} 2 & 4 \\ 3 & 3 \end{pmatrix}$

This matches option a.

Final Answer:

The correct answer is a.

Would you like more details on any part of the solution? Here are 5 related questions for further practice:

1. How do you multiply a matrix by a scalar?
2. What are the properties of matrix addition?
3. How would the solution change if the given matrix was 3x3 instead of 2x2?
4. Can you derive the inverse of matrix $Q$?
5. How do you solve a system of equations using matrices?

Tip: When solving matrix equations, always perform operations element-wise and verify your results step-by-step.