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The given problem appears to involve matrix equations, where we are tasked with finding the value of \( C \) that satisfies the matrix equation.

The equation is:

\[
\left[ \begin{array}{cc} 4 & b-2 \\ c-2b & 4 \end{array} \right] + \left[ \begin{array}{cc} -ac & -1 \\ 8 & -6 \end{array} \right] = 2 \left[ \begin{array}{cc} -10 & 4 \\ -4 & -1 \end{array} \right]
\]

We will first solve the right-hand side of the equation by multiplying the matrix by 2:

\[
2 \times \left[ \begin{array}{cc} -10 & 4 \\ -4 & -1 \end{array} \right] = \left[ \begin{array}{cc} -20 & 8 \\ -8 & -2 \end{array} \right]
\]

Now, we can rewrite the equation as:

\[
\left[ \begin{array}{cc} 4 & b-2 \\ c-2b & 4 \end{array} \right] + \left[ \begin{array}{cc} -ac & -1 \\ 8 & -6 \end{array} \right] = \left[ \begin{array}{cc} -20 & 8 \\ -8 & -2 \end{array} \right]
\]

Next, add the two matrices on the left-hand side:

\[
\left[ \begin{array}{cc} 4 + (-ac) & (b-2) + (-1) \\ (c-2b) + 8 & 4 + (-6) \end{array} \right] = \left[ \begin{array}{cc} -20 & 8 \\ -8 & -2 \end{array} \right]
\]

Simplify this to:

\[
\left[ \begin{array}{cc} 4 - ac & b - 3 \\ c - 2b + 8 & -2 \end{array} \right] = \left[ \begin{array}{cc} -20 & 8 \\ -8 & -2 \end{array} \right]
\]

Now, we can equate corresponding elements from both sides:

1. \( 4 - ac = -20 \)
2. \( b - 3 = 8 \)
3. \( c - 2b + 8 = -8 \)
4. \( -2 = -2 \) (this is true, so it provides no new information)

From the second equation:

\[
b - 3 = 8 \implies b = 11
\]

Substitute \( b = 11 \) into the third equation:

\[
c - 2(11) + 8 = -8 \implies c - 22 + 8 = -8 \implies c - 14 = -8 \implies c = 6
\]

Finally, check the first equation. Substitute \( c = 6 \) and \( b = 11 \):

\[
4 - a(6) = -20 \implies 4 - 6a = -20 \implies -6a = -24 \implies a = 4
\]

Therefore, the value of \( C \) is \( \boxed{6} \).

Would you like further clarification, or have any other questions?

Here are 5 related questions you can explore:

1. How do you solve a system of matrix equations?
2. How does scalar multiplication affect matrix operations?
3. Can you explain how to find the inverse of a matrix?
4. What are the properties of matrix addition and multiplication?
5. How do you interpret determinant values of matrices?

**Tip:** When solving matrix equations, always carefully align corresponding elements from both sides for comparison.

Find the value of C that satisfies the matrix equation: [ [4, b-2], [c-2b, 4] ] + [ [-ac, -1], [8, -6] ] = 2 [ [-10, 4], [-4, -1] ]

Learn how to solve a matrix equation by finding the value of C in a system that includes matrix addition and scalar multiplication. This solution walks through the necessary steps of matrix arithmetic and comparison.

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