Which of the following matrices is equal to `4[(-1,2),(0,-4)]`?

Options

A. `[(-4,-8)]`
B. `[(4),(-16)]`
C. `[(3,6),(4,0)]`
D. `[(-4,8),(0,-16)]`

Answer

To solve the problem of finding which matrix is equal to \( 4 \begin{pmatrix} -1 & 2 \\ 0 & -4 \end{pmatrix} \), we will perform scalar multiplication of the matrix by 4. Here are the steps:

### Step 1: Identify the matrix
The given matrix is:
\[ \begin{pmatrix} -1 & 2 \\ 0 & -4 \end{pmatrix} \]

### Step 2: Perform scalar multiplication
We need to multiply each element of the matrix by 4. This means we will calculate:
- \( 4 \times -1 \)
- \( 4 \times 2 \)
- \( 4 \times 0 \)
- \( 4 \times -4 \)

### Step 3: Calculate each element
1. \( 4 \times -1 = -4 \)
2. \( 4 \times 2 = 8 \)
3. \( 4 \times 0 = 0 \)
4. \( 4 \times -4 = -16 \)

### Step 4: Form the new matrix
After performing the calculations, we can form the new matrix:
\[ \begin{pmatrix} -4 & 8 \\ 0 & -16 \end{pmatrix} \]

### Step 5: Compare with the options
Now we compare this resulting matrix with the given options:
- A: \( \begin{pmatrix} -4 & -8 \end{pmatrix} \)
- B: \( \begin{pmatrix} 4 \\ -16 \end{pmatrix} \)
- C: \( \begin{pmatrix} 3 & 6 \\ 4 & 0 \end{pmatrix} \)
- D: \( \begin{pmatrix} -4 & 8 \\ 0 & -16 \end{pmatrix} \)

From the comparison, we see that the resulting matrix \( \begin{pmatrix} -4 & 8 \\ 0 & -16 \end{pmatrix} \) matches option D.

### Conclusion
Thus, the matrix that is equal to \( 4 \begin{pmatrix} -1 & 2 \\ 0 & -4 \end{pmatrix} \) is:
\[ \begin{pmatrix} -4 & 8 \\ 0 & -16 \end{pmatrix} \]
which corresponds to option D.