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If xyz=m and det p=|{:(x,y,z),(z,x,y),(y...

If xyz=m and det `p=|{:(x,y,z),(z,x,y),(y,z,x):}|` , where p is an orthogonal matrix.
The value of `x^(-1)+y^(-1)+z^(-1)` is

A

`+-` m

B

` +- 1`

C

0

D

`m^(2) `

Text Solution

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The correct Answer is:
To solve the problem, we need to find the value of \( x^{-1} + y^{-1} + z^{-1} \) given that \( xyz = m \) and the determinant of matrix \( P \) is defined as: \[ P = \begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix} \] Since \( P \) is an orthogonal matrix, we know that: \[ P P^T = I \] where \( I \) is the identity matrix. This means that the rows (and columns) of \( P \) are orthonormal vectors. ### Step 1: Calculate \( P^T \) The transpose of matrix \( P \) is: \[ P^T = \begin{pmatrix} x & z & y \\ y & x & z \\ z & y & x \end{pmatrix} \] ### Step 2: Calculate \( P P^T \) Now, we compute the product \( P P^T \): \[ P P^T = \begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix} \begin{pmatrix} x & z & y \\ y & x & z \\ z & y & x \end{pmatrix} \] Calculating the elements of this product: - The (1,1) entry is \( x^2 + y^2 + z^2 \) - The (1,2) entry is \( xy + yz + zx \) - The (1,3) entry is \( xy + yz + zx \) - The (2,1) entry is \( xy + yz + zx \) - The (2,2) entry is \( z^2 + x^2 + y^2 \) - The (2,3) entry is \( xy + yz + zx \) - The (3,1) entry is \( xy + yz + zx \) - The (3,2) entry is \( xy + yz + zx \) - The (3,3) entry is \( y^2 + z^2 + x^2 \) Thus, we have: \[ P P^T = \begin{pmatrix} x^2 + y^2 + z^2 & xy + yz + zx & xy + yz + zx \\ xy + yz + zx & z^2 + x^2 + y^2 & xy + yz + zx \\ xy + yz + zx & xy + yz + zx & y^2 + z^2 + x^2 \end{pmatrix} \] ### Step 3: Set equal to the identity matrix Since \( P P^T = I \), we can compare the entries: 1. \( x^2 + y^2 + z^2 = 1 \) 2. \( xy + yz + zx = 0 \) ### Step 4: Find \( x^{-1} + y^{-1} + z^{-1} \) We know that: \[ x^{-1} + y^{-1} + z^{-1} = \frac{yz + zx + xy}{xyz} \] Given that \( xy + yz + zx = 0 \) and \( xyz = m \): \[ x^{-1} + y^{-1} + z^{-1} = \frac{0}{m} = 0 \] ### Final Answer Thus, the value of \( x^{-1} + y^{-1} + z^{-1} \) is: \[ \boxed{0} \]
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