If `A=[{:(11,7),(-13,17):}]` then adj(adj A) is

Options

A. `[{:(17,-7),(13,11):}]`
B. `[{:(11,7),(-13,17):}]`
C. `[{:(-17,7),(13,11):}]`
D. `[{:(-11,7),(-13,17):}]`

Answer

To find \( \text{adj}(\text{adj} A) \) for the matrix \( A = \begin{pmatrix} 11 & 7 \\ -13 & 17 \end{pmatrix} \), we can use the property of determinants that states:

\[ \text{adj}(\text{adj} A) = \det(A) A \]

### Step 1: Calculate the determinant of \( A \)

The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ \det(A) = ad - bc \]

For our matrix \( A \):

- \( a = 11 \)
- \( b = 7 \)
- \( c = -13 \)
- \( d = 17 \)

Calculating the determinant:

\[ \det(A) = (11)(17) - (7)(-13) = 187 + 91 = 278 \]

### Step 2: Find \( \text{adj} A \)

The adjoint of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Applying this to our matrix \( A \):

\[ \text{adj}(A) = \begin{pmatrix} 17 & -7 \\ 13 & 11 \end{pmatrix} \]

### Step 3: Find \( \text{adj}(\text{adj} A) \)

Now we need to find the adjoint of \( \text{adj} A \):

\[ \text{adj}(\text{adj} A) = \text{adj}\left(\begin{pmatrix} 17 & -7 \\ 13 & 11 \end{pmatrix}\right) \]

Using the adjoint formula again:

\[ \text{adj}(\text{adj} A) = \begin{pmatrix} 11 & 7 \\ -13 & 17 \end{pmatrix} \]

### Step 4: Final Result

Thus, we find that:

\[ \text{adj}(\text{adj} A) = A \]

### Conclusion

The final answer is:

\[ \text{adj}(\text{adj} A) = \begin{pmatrix} 11 & 7 \\ -13 & 17 \end{pmatrix} \]