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The problem asks to determine \(|a - d|\) from a given matrix equation. We are given the matrix \( A = \begin{pmatrix} a & -3 \\ 1 & d \end{pmatrix} \), and we know that \( A = A^{-1} \).

### Step-by-step Solution:
1. **Matrix Inversion:**
   The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated by:
   \[
   A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
   \]
   So for \( A = \begin{pmatrix} a & -3 \\ 1 & d \end{pmatrix} \), we have:
   \[
   A^{-1} = \frac{1}{ad - (-3)} \begin{pmatrix} d & 3 \\ -1 & a \end{pmatrix}
   \]

2. **Equating \( A \) and \( A^{-1} \):**
   Since \( A = A^{-1} \), the matrices must be equal element by element:
   \[
   \frac{1}{ad + 3} \begin{pmatrix} d & 3 \\ -1 & a \end{pmatrix} = \begin{pmatrix} a & -3 \\ 1 & d \end{pmatrix}
   \]
   This leads to the system of equations:
   - \(\frac{d}{ad + 3} = a\)
   - \(\frac{3}{ad + 3} = -3\)
   - \(\frac{-1}{ad + 3} = 1\)
   - \(\frac{a}{ad + 3} = d\)

3. **Solving the system:**
   Let's start with the second equation:
   \[
   \frac{3}{ad + 3} = -3 \quad \Rightarrow \quad 3 = -3(ad + 3) \quad \Rightarrow \quad 3 = -3ad - 9
   \]
   Simplifying:
   \[
   3ad = -12 \quad \Rightarrow \quad ad = -4
   \]

4. **Substitute into the first equation:**
   Using \( ad = -4 \) in the first equation:
   \[
   \frac{d}{-4 + 3} = a \quad \Rightarrow \quad \frac{d}{-1} = a \quad \Rightarrow \quad a = -d
   \]

5. **Final calculation:**
   Since \( a = -d \), we want to find \( |a - d| \):
   \[
   |a - d| = |-d - d| = |-2d| = 2|d|
   \]
   We know from \( ad = -4 \) that \( (-d)d = -4 \), which simplifies to \( d^2 = 4 \), so \( d = 2 \) or \( d = -2 \).

6. **Conclusion:**
   Whether \( d = 2 \) or \( d = -2 \), we have \( |a - d| = 4 \).

Thus, the value of \( |a - d| \) is **4**, so the correct answer is:

\[
\boxed{E}
\]

Would you like more details on any of the steps? Here are five related questions to explore further:

1. How do you compute the determinant of a 2x2 matrix?
2. What is the general method for finding the inverse of a matrix?
3. Can a matrix always be equal to its inverse? Why or why not?
4. What are the conditions for a matrix to be invertible?
5. How do we interpret the absolute value in matrix-related problems?

**Tip:** When matrices are involved, always check the determinant first to ensure it's invertible!

Diketahui A = [[a, -3], [1, d]]. Jika A = A^-1, nilai |a - d| adalah...

This problem involves finding |a - d| for a matrix where A = A^-1. Learn how to solve this using matrix inversion and determinant properties, suitable for Grades 10-12.

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