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.