If `a_(1)^(2)+a_(2)^(2)+a_(3)^(2)=1, b_(1)^(2)+b_(2)^(2)+b_(3)^(2)=4, c_(1)^(2)+c_(2)^(2)+c_3^(2)=9, a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3)=0, a_(1)c_(1)+a_(2)c_(2)+a_(3)c_(3)=0, b_(1)c_(1)+b_(2)c_(2)+b_(3)c_(3)=0 and A[(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))]`, then `|A|^(4)` is equal to

Options

A. 36
B. 49
C. 1296
D. 216

Answer

To solve the problem, we need to evaluate the determinant \( |A|^4 \) where \( A \) is defined as:

\[ A = \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix} \]

Given the conditions:

1. \( a_1^2 + a_2^2 + a_3^2 = 1 \)
2. \( b_1^2 + b_2^2 + b_3^2 = 4 \)
3. \( c_1^2 + c_2^2 + c_3^2 = 9 \)
4. \( a_1b_1 + a_2b_2 + a_3b_3 = 0 \)
5. \( a_1c_1 + a_2c_2 + a_3c_3 = 0 \)
6. \( b_1c_1 + b_2c_2 + b_3c_3 = 0 \)

### Step 1: Construct the matrix \( A \)

The matrix \( A \) is constructed from the vectors \( (a_1, a_2, a_3) \), \( (b_1, b_2, b_3) \), and \( (c_1, c_2, c_3) \).

### Step 2: Calculate the determinant \( |A| \)

Using the properties of determinants and the given conditions, we can express the determinant \( |A| \) as follows:

\[ |A|^2 = (a_1^2 + a_2^2 + a_3^2)(b_1^2 + b_2^2 + b_3^2)(c_1^2 + c_2^2 + c_3^2) - (a_1b_1 + a_2b_2 + a_3b_3)^2 - (a_1c_1 + a_2c_2 + a_3c_3)^2 - (b_1c_1 + b_2c_2 + b_3c_3)^2 \]

### Step 3: Substitute the known values

From the problem statement, we have:

- \( a_1^2 + a_2^2 + a_3^2 = 1 \)
- \( b_1^2 + b_2^2 + b_3^2 = 4 \)
- \( c_1^2 + c_2^2 + c_3^2 = 9 \)
- \( a_1b_1 + a_2b_2 + a_3b_3 = 0 \)
- \( a_1c_1 + a_2c_2 + a_3c_3 = 0 \)
- \( b_1c_1 + b_2c_2 + b_3c_3 = 0 \)

Substituting these values into the determinant formula gives:

\[ |A|^2 = (1)(4)(9) - 0^2 - 0^2 - 0^2 = 36 \]

### Step 4: Calculate \( |A| \)

Taking the square root to find \( |A| \):

\[ |A| = \sqrt{36} = 6 \]

### Step 5: Calculate \( |A|^4 \)

Now we compute \( |A|^4 \):

\[ |A|^4 = (|A|^2)^2 = 36^2 = 1296 \]

### Final Answer

Thus, the value of \( |A|^4 \) is:

\[ \boxed{1296} \]