If ` [{:( 1,2,3),(1,3,5),(1,5,12):}]` then adj (adj A) is

Options

A. ` [{:( 3,3,3),(6,9,15),(9,15,36):}]`
B. ` [{:( 1,2,3),(1,3,5),(1,5,12):}]`
C. ` [{:( 3,6,9),(3,9,15),( 3,15,36):}]`
D. none of these

Answer

To find the adjoint of the adjoint of the matrix \( A \), we can use the property that states:

\[ \text{adj}(\text{adj}(A)) = (\det A)^{n-2} A \]

where \( n \) is the order of the matrix \( A \).

### Step-by-Step Solution:

1. **Identify the matrix \( A \)**:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 1 & 5 & 12 \end{pmatrix} \]

2. **Determine the order of the matrix \( A \)**:
The matrix \( A \) is a \( 3 \times 3 \) matrix, so \( n = 3 \).

3. **Calculate the determinant of \( A \)**:
We will calculate \( \det(A) \) using the formula for the determinant of a \( 3 \times 3 \) matrix:
\[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix elements are:
\[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
For our matrix:
\[ a = 1, b = 2, c = 3, d = 1, e = 3, f = 5, g = 1, h = 5, i = 12 \]
Thus,
\[ \det(A) = 1(3 \cdot 12 - 5 \cdot 5) - 2(1 \cdot 12 - 5 \cdot 1) + 3(1 \cdot 5 - 3 \cdot 1) \]
Calculating each term:
- First term: \( 1(36 - 25) = 1 \cdot 11 = 11 \)
- Second term: \( -2(12 - 5) = -2 \cdot 7 = -14 \)
- Third term: \( 3(5 - 3) = 3 \cdot 2 = 6 \)

Now summing these:
\[ \det(A) = 11 - 14 + 6 = 3 \]

4. **Use the property of the adjoint**:
Now, we can find \( \text{adj}(\text{adj}(A)) \) using the formula:
\[ \text{adj}(\text{adj}(A)) = (\det A)^{n-2} A \]
Here, \( n = 3 \), so \( n - 2 = 1 \). Thus:
\[ \text{adj}(\text{adj}(A)) = (\det A)^{1} A = 3A \]

5. **Calculate \( 3A \)**:
Multiply each element of \( A \) by 3:
\[ 3A = 3 \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 1 & 5 & 12 \end{pmatrix} = \begin{pmatrix} 3 & 6 & 9 \\ 3 & 9 & 15 \\ 3 & 15 & 36 \end{pmatrix} \]

### Final Result:
\[ \text{adj}(\text{adj}(A)) = \begin{pmatrix} 3 & 6 & 9 \\ 3 & 9 & 15 \\ 3 & 15 & 36 \end{pmatrix} \]