Let `A=[(1,1,1),(1,-1,0),(0,1,-1)], A_(1)` be a matrix formed by the cofactors of the elements of the matrix A and `A_(2)` be a matrix formed by the cofactors of the elements of matrix `A_(1)`. Similarly, If `A_(10)` be a matrrix formed by the cofactors of the elements of matrix `A_(9)`, then the value of `|A_(10)|` is

Options

A. `3^(10)`
B. `3^(20)`
C. 9
D. `3^(1024)`

Answer

To solve the problem, we need to find the value of \(|A_{10}|\) where \(A\) is given as:

\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{pmatrix} \]

### Step 1: Calculate the Determinant of Matrix \(A\)

To find \(|A|\), we can use the formula for the determinant of a \(3 \times 3\) matrix:

\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \(A\):

\[ |A| = 1 \cdot ((-1) \cdot (-1) - 0 \cdot 1) - 1 \cdot (1 \cdot (-1) - 0 \cdot 0) + 1 \cdot (1 \cdot 1 - (-1) \cdot 0) \]

Calculating this step by step:

1. Calculate \((-1) \cdot (-1) - 0 \cdot 1 = 1 - 0 = 1\)
2. Calculate \(1 \cdot (-1) - 0 \cdot 0 = -1 - 0 = -1\)
3. Calculate \(1 \cdot 1 - (-1) \cdot 0 = 1 - 0 = 1\)

Putting it all together:

\[ |A| = 1 \cdot 1 - 1 \cdot (-1) + 1 \cdot 1 = 1 + 1 + 1 = 3 \]

### Step 2: Determine the Determinant of \(A_{1}\)

The determinant of the cofactor matrix \(A_{1}\) is given by the property:

\[ |A_{1}| = |A|^2 \]

Thus,

\[ |A_{1}| = 3^2 = 9 \]

### Step 3: Determine the Determinant of \(A_{2}\)

Similarly, the determinant of \(A_{2}\) is:

\[ |A_{2}| = |A_{1}|^2 = 9^2 = 81 \]

### Step 4: Generalize for \(A_{n}\)

Following the same pattern, we can see that:

\[ |A_{n}| = |A_{n-1}|^2 \]

Continuing this process, we can derive:

- \(|A_{3}| = |A_{2}|^2 = 81^2 = 6561\)
- \(|A_{4}| = |A_{3}|^2 = 6561^2 = 43046721\)
- \(|A_{5}| = |A_{4}|^2\)
- \(|A_{6}| = |A_{5}|^2\)
- \(|A_{7}| = |A_{6}|^2\)
- \(|A_{8}| = |A_{7}|^2\)
- \(|A_{9}| = |A_{8}|^2\)
- \(|A_{10}| = |A_{9}|^2\)

### Step 5: Calculate \(|A_{10}|\)

From the pattern, we can express \(|A_{n}|\) as:

\[ |A_{n}| = |A|^{2^n} \]

Thus,

\[ |A_{10}| = |A|^{2^{10}} = 3^{2^{10}} = 3^{1024} \]

### Final Answer

The value of \(|A_{10}|\) is:

\[ \boxed{3^{1024}} \]