If a(1)^(2)+a(2)^(2)+a(3)^(2)=1, b(1)^(2...

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

1296

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To solve the problem, we need to find the determinant of the matrix \( A \) given the conditions provided. Let's break it down step by step. ### Step 1: Define the Matrix \( A \) The matrix \( A \) is defined as: \[ A = \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix} \] ### Step 2: Use the Given Conditions We have the following 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 3: Calculate \( A^2 \) To find \( A^2 \), we multiply the matrix \( A \) by itself: \[ A^2 = A \cdot A = \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix} \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix} \] Calculating the entries of \( A^2 \): - The first row, first column: \[ a_1^2 + a_2^2 + a_3^2 = 1 \] - The first row, second column: \[ a_1a_2 + a_2b_2 + a_3b_3 = 0 \quad (\text{given}) \] - The first row, third column: \[ a_1c_1 + a_2c_2 + a_3c_3 = 0 \quad (\text{given}) \] Continuing this process for the other rows, we find: \[ A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 9 \end{pmatrix} \] ### Step 4: Calculate the Determinant of \( A^2 \) The determinant of a diagonal matrix is the product of its diagonal entries: \[ |A^2| = 1 \cdot 4 \cdot 9 = 36 \] ### Step 5: Calculate \( |A|^4 \) Since \( |A^2| = |A|^2 \), we have: \[ |A|^2 = 36 \] Taking the square root gives us: \[ |A| = 6 \] Now, we need \( |A|^4 \): \[ |A|^4 = (|A|^2)^2 = 36^2 = 1296 \] ### Final Answer Thus, the value of \( |A|^4 \) is: \[ \boxed{1296} \]

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