The problem provided involves matrix operations. Here's a breakdown of the question from the image:

1. We are given the matrices:

   • $A = \begin{pmatrix} 1 & 2 \\ -4 & 3 \end{pmatrix}$
   • $B = \begin{pmatrix} -3 & 2 \\ -1 & 5 \end{pmatrix}$
   • $C = \begin{pmatrix} -3 & -4 \\ 1 & 2 \end{pmatrix}$

2. We also have the equation: $A^T + 3B^T - 3C - 2X = 0$ Where $A^T$ and $B^T$ represent the transpose of matrices $A$ and $B$, respectively, and $X$ is an unknown matrix that we need to find.

Step-by-Step Approach:

1. Transpose of Matrices $A$ and $B$:

   • The transpose of matrix $A$ ($A^T$) is: $A^T = \begin{pmatrix} 1 & -4 \\ 2 & 3 \end{pmatrix}$
   • The transpose of matrix $B$ ($B^T$) is: $B^T = \begin{pmatrix} -3 & -1 \\ 2 & 5 \end{pmatrix}$

2. Matrix Multiplications:

   • Calculate $3B^T$: $3B^T = 3 \times \begin{pmatrix} -3 & -1 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} -9 & -3 \\ 6 & 15 \end{pmatrix}$
   • Calculate $3C$: $3C = 3 \times \begin{pmatrix} -3 & -4 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} -9 & -12 \\ 3 & 6 \end{pmatrix}$

3. Substitute into the Equation: The original equation is: $A^T + 3B^T - 3C - 2X = 0$ Substituting the values: $\begin{pmatrix} 1 & -4 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} -9 & -3 \\ 6 & 15 \end{pmatrix} - \begin{pmatrix} -9 & -12 \\ 3 & 6 \end{pmatrix} - 2X = 0$

4. Simplify the Left Side: Combine the matrices: $\begin{pmatrix} 1 & -4 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} -9 & -3 \\ 6 & 15 \end{pmatrix} = \begin{pmatrix} -8 & -7 \\ 8 & 18 \end{pmatrix}$ Now subtract $3C$: $\begin{pmatrix} -8 & -7 \\ 8 & 18 \end{pmatrix} - \begin{pmatrix} -9 & -12 \\ 3 & 6 \end{pmatrix} = \begin{pmatrix} 1 & 5 \\ 5 & 12 \end{pmatrix}$ So, the equation becomes: $\begin{pmatrix} 1 & 5 \\ 5 & 12 \end{pmatrix} - 2X = 0$

5. Solve for $X$: Rearrange the equation: $2X = \begin{pmatrix} 1 & 5 \\ 5 & 12 \end{pmatrix}$ Divide both sides by 2: $X = \frac{1}{2} \times \begin{pmatrix} 1 & 5 \\ 5 & 12 \end{pmatrix} = \begin{pmatrix} 0.5 & 2.5 \\ 2.5 & 6 \end{pmatrix}$

Thus, the matrix $X$ is: $X = \begin{pmatrix} 0.5 & 2.5 \\ 2.5 & 6 \end{pmatrix}$

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. What is the process for finding the transpose of a matrix?
2. How would you perform matrix multiplication and addition?
3. How do you handle matrix equations like the one in the problem?
4. What properties of matrices are important when working with transposes?
5. How does scalar multiplication work in matrix operations?

Tip: When solving matrix equations, always work step by step, simplifying matrices at each stage to avoid errors.