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The question you're asking deals with matrices and seems to be asking for the value of \( d \), based on the given matrix equations. Let's break it down:

### Problem Setup:
You are given three matrices \( A \), \( B \), and \( C \):

\[
A = \begin{pmatrix} 1 & a + 6 \\ 6 & c \end{pmatrix}, \quad B = \begin{pmatrix} a - 1 & 0 \\ -c & d \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}
\]

The equation provided is:
\[
A + B = C
\]

We need to find the value of \( d \).

### Step 1: Add the matrices \( A \) and \( B \)
\[
A + B = \begin{pmatrix} 1 & a + 6 \\ 6 & c \end{pmatrix} + \begin{pmatrix} a - 1 & 0 \\ -c & d \end{pmatrix}
\]

\[
A + B = \begin{pmatrix} (1 + (a - 1)) & ((a + 6) + 0) \\ (6 + (-c)) & (c + d) \end{pmatrix}
\]

Simplifying this:

\[
A + B = \begin{pmatrix} a & a + 6 \\ 6 - c & c + d \end{pmatrix}
\]

### Step 2: Set the resulting matrix equal to matrix \( C \):
\[
\begin{pmatrix} a & a + 6 \\ 6 - c & c + d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}
\]

This gives us a system of equations by comparing corresponding elements:

1. \( a = 1 \)
2. \( a + 6 = 0 \)
3. \( 6 - c = 1 \)
4. \( c + d = 1 \)

### Step 3: Solve the system of equations
From the second equation:
\[
a + 6 = 0 \implies a = -6
\]

From the third equation:
\[
6 - c = 1 \implies c = 5
\]

Finally, using the value of \( c \) in the fourth equation:
\[
c + d = 1 \implies 5 + d = 1 \implies d = 1 - 5 = -4
\]

### Final Answer:
The value of \( d \) is \( \boxed{-4} \).

---

Would you like further clarification or details on any step?

Here are 5 related questions for you to explore:
1. What happens if the matrix \( B \) has different dimensions than \( A \)?
2. How can we use matrix multiplication instead of addition in a similar problem?
3. What if the matrix \( C \) had a different dimension (e.g., a 3x3 matrix)?
4. How does matrix inversion work, and can it be applied here?
5. Can you find the determinant of matrix \( A \) and \( B \)?

**Tip:** Always make sure the matrices are of the same dimension when adding or subtracting them!

If A + B = C, find the value of d. Matrices A, B, and C are given.

This matrix addition problem asks for the value of d when A + B = C. The matrices A, B, and C are given, and the solution involves using matrix addition and solving a system of equations. Suitable for high school students in Grades 10-12.

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