Let's break down the tasks in the uploaded control work:

1. Matrix Operations: a) $3A + B^T$: You need to calculate $3A$, where $A$ is given, and then compute the transpose of matrix $B$, denoted as $B^T$, and finally add these two matrices together.

   b) $B \cdot A - 3E$: This involves multiplying matrices $B$ and $A$, and then subtracting $3E$, where $E$ is likely the identity matrix (if not otherwise specified).

2. Determinants: a) Calculate the determinant of the matrix:

   -3 & 4 \\ 5 & 6 \end{pmatrix}$$ b) Calculate the determinant of the matrix: $$\begin{pmatrix} 2 & 0 & -1 \\ 1 & 2 & 0 \\ 4 & -1 & 0 \end{pmatrix}$$

3. Equation Solution: Solve for $X$ in the matrix equation: $X \cdot \begin{pmatrix} 1 & -2 & 3 \\ 3 & 1 & 0 \\ 4 & 3 & 5 \end{pmatrix} = \begin{pmatrix} -15 & 17 & 1 \end{pmatrix}$

4. System of Equations: Solve the system:

   x_1 + 2x_2 - 3x_3 &= 10 \\ -2x_1 - x_2 + 2x_3 &= -10 \\ x_1 - 3x_3 &= 6 \end{aligned}$$ Determine if the system is balanced.

Would you like me to start solving these problems step-by-step? Feel free to specify which problem you'd like to begin with!