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The problem given involves two matrices, \( A \) and \( B \), and asks to find the value of \( a + b \) given that matrix \( A \) is equal to the transpose of matrix \( B \). Let's break down the problem and solve it.

We are given:
\[
A = \begin{pmatrix} 10 & -9 \\ 7 & -b \end{pmatrix}, \quad B = \begin{pmatrix} a & -2b \\ -9 & 7 \end{pmatrix}
\]
The condition provided is that \( A = B^T \), where \( B^T \) is the transpose of \( B \). The transpose of matrix \( B \) is obtained by swapping its rows and columns:
\[
B^T = \begin{pmatrix} a & -9 \\ -2b & 7 \end{pmatrix}
\]

Now, equating \( A \) and \( B^T \):
\[
\begin{pmatrix} 10 & -9 \\ 7 & -b \end{pmatrix} = \begin{pmatrix} a & -9 \\ -2b & 7 \end{pmatrix}
\]

From this, we can compare corresponding elements of the matrices:

1. \( 10 = a \)  (from the element in the first row and first column)
2. \( -9 = -9 \)  (this holds true automatically)
3. \( 7 = 7 \)  (this also holds true)
4. \( -b = -2b \)

From the equation \( -b = -2b \), we get:
\[
b = 0
\]

Substituting \( b = 0 \) into \( 10 = a \), we have:
\[
a = 10
\]

Thus, the value of \( a + b \) is:
\[
a + b = 10 + 0 = 10
\]

The value of \( a + b \) is **10**.

Would you like a more detailed explanation, or do you have any further questions?

Here are five related questions to explore the topic further:
1. How do you calculate the transpose of a matrix?
2. What properties of a matrix remain unchanged after taking the transpose?
3. Can two non-identical matrices ever be transposes of each other?
4. What is the significance of symmetric matrices in relation to transposes?
5. How does the determinant of a matrix compare with that of its transpose?

**Tip**: When comparing matrices, always ensure to equate the corresponding elements and solve for any variables.

Diketahui matriks A = (10  -9; 7  -b) dan B = (a  -2b; -9  7). Jika matriks A = B^T, maka nilai a + b adalah ...

This problem involves matrices A and B, and asks to find the value of a + b given that A equals the transpose of B. The solution uses matrix transposition and equality of matrices, suitable for high school students (Grades 10-12).

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