The given problem shows two matrices $A$ and $B$, and we are required to find the value of $\frac{1}{a} + \frac{1}{b}$ under the condition $A^T = B$.

Matrix $A$:

$A = \begin{pmatrix} a+b & 0 \\ 2 & 9 \end{pmatrix}$

Matrix $B$:

$B = \begin{pmatrix} 8 & 0 \\ 2 & ab \end{pmatrix}$

Given Condition:

$A^T = B$ This means the transpose of matrix $A$ must equal matrix $B$.

Transpose of $A$:

The transpose of matrix $A$, denoted as $A^T$, is: $A^T = \begin{pmatrix} a+b & 2 \\ 0 & 9 \end{pmatrix}$

For $A^T = B$, the corresponding elements of the two matrices must be equal: $\begin{pmatrix} a+b & 2 \\ 0 & 9 \end{pmatrix} = \begin{pmatrix} 8 & 0 \\ 2 & ab \end{pmatrix}$

From this equality, we can derive the following relationships:

1. $a + b = 8$ (from the top-left elements),
2. $2 = 2$ (this is already satisfied),
3. $ab = 9$ (from the bottom-right elements).

Now, we need to find the value of $\frac{1}{a} + \frac{1}{b}$. Using the identity:

$\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$

Substitute the values we found:

• $a + b = 8$
• $ab = 9$

Thus: $\frac{1}{a} + \frac{1}{b} = \frac{8}{9}$

The correct answer is $\frac{8}{9}$, which corresponds to the last option.

Answer: $\frac{8}{9}$

Would you like further details or have any other questions?

Here are 5 related questions to expand on this:

1. How do you find the inverse of a 2x2 matrix?
2. What is the geometric interpretation of the transpose of a matrix?
3. Can you explain the relationship between determinant and area in 2x2 matrices?
4. How can matrix equations be solved using row reduction techniques?
5. What are the properties of symmetric matrices?

Tip: When solving systems involving matrices, always check if the given condition (like $A^T = B$) simplifies the problem before performing complex calculations!