Let's break down each matrix operation given and solve them individually.

1. Matrix Definitions:

   • $A = \begin{bmatrix} 4 & 1 \\ -3 & 0 \end{bmatrix}$
   • $B = \begin{bmatrix} -2 & 3 \\ -5 & 1 \end{bmatrix}$
   • $C = \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}$

2. Calculating $3A$: To find $3A$, multiply each element of matrix $A$ by 3: $3A = 3 \times \begin{bmatrix} 4 & 1 \\ -3 & 0 \end{bmatrix} = \begin{bmatrix} 12 & 3 \\ -9 & 0 \end{bmatrix}$

3. Calculating $A + B$: To add $A$ and $B$, add the corresponding elements: $A + B = \begin{bmatrix} 4 & 1 \\ -3 & 0 \end{bmatrix} + \begin{bmatrix} -2 & 3 \\ -5 & 1 \end{bmatrix} = \begin{bmatrix} 4 + (-2) & 1 + 3 \\ -3 + (-5) & 0 + 1 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ -8 & 1 \end{bmatrix}$

4. Calculating $BC$: To multiply matrices $B$ and $C$, perform the matrix multiplication: $BC = \begin{bmatrix} -2 & 3 \\ -5 & 1 \end{bmatrix} \times \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}$ Calculating each element:

   • Top-left: $(-2 \times 5) + (3 \times 2) = -10 + 6 = -4$
   • Top-right: $(-2 \times -1) + (3 \times 3) = 2 + 9 = 11$
   • Bottom-left: $(-5 \times 5) + (1 \times 2) = -25 + 2 = -23$
   • Bottom-right: $(-5 \times -1) + (1 \times 3) = 5 + 3 = 8$

   So, $BC = \begin{bmatrix} -4 & 11 \\ -23 & 8 \end{bmatrix}$

Summary of Results:

• $3A = \begin{bmatrix} 12 & 3 \\ -9 & 0 \end{bmatrix}$
• $A + B = \begin{bmatrix} 2 & 4 \\ -8 & 1 \end{bmatrix}$
• $BC = \begin{bmatrix} -4 & 11 \\ -23 & 8 \end{bmatrix}$

Now, you can match these results with the provided options in your answer choices.

Would you like further explanations on any part of this? Here are a few questions to expand on the concept:

1. How is scalar multiplication of matrices performed?
2. What are the conditions for matrix addition and subtraction?
3. Can all matrices be multiplied with each other? If not, what are the conditions?
4. What happens when you multiply a matrix by an identity matrix?
5. How do these operations apply in real-world scenarios?

Tip: When multiplying matrices, remember that the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.