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To find which of the given matrices equals the linear combination \(A - 3B\), let's perform the calculation step by step.

Given matrices:
\[
A = \begin{bmatrix} -4 & 1 \\ 0 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}
\]

We want to find \( A - 3B \).

1. Multiply matrix \( B \) by 3:
   \[
   3B = 3 \times \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} 6 & 3 \\ -3 & 9 \end{bmatrix}
   \]

2. Subtract \( 3B \) from \( A \):
   \[
   A - 3B = \begin{bmatrix} -4 & 1 \\ 0 & 6 \end{bmatrix} - \begin{bmatrix} 6 & 3 \\ -3 & 9 \end{bmatrix} = \begin{bmatrix} -4 - 6 & 1 - 3 \\ 0 - (-3) & 6 - 9 \end{bmatrix} = \begin{bmatrix} -10 & -2 \\ 3 & -3 \end{bmatrix}
   \]

Now, let's check the options:
- The correct option that matches \(\begin{bmatrix} -10 & -2 \\ 3 & -3 \end{bmatrix}\) is the fifth option: \(\begin{bmatrix} -10 & -2 \\ 3 & -3 \end{bmatrix}\).

Therefore, the answer is the fifth matrix: \(\boxed{\begin{bmatrix} -10 & -2 \\ 3 & -3 \end{bmatrix}}\).

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:
1. How would you compute the combination \(2A + 4B\)?
2. What is the determinant of matrix \(A\)?
3. Can you calculate the inverse of matrix \(B\)?
4. How would you find the trace of matrix \(A - 3B\)?
5. What are the eigenvalues of matrix \(A\)?

**Tip:** Always double-check your arithmetic calculations when performing matrix operations to avoid simple errors.

Which of the following matrices equals the linear combination A - 3B?

Explore how to calculate the linear combination A - 3B using matrix subtraction and scalar multiplication. This problem demonstrates fundamental concepts in linear algebra, suitable for college-level students.

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