If xyz=m and det `p=|{:(x,y,z),(z,x,y),(y,z,x):}|` , where p is an orthogonal matrix.
If y=x=z, then z is equal to

Options

A. `pm 1root/3`
B. `pm 5/3`
C. `pm 2/3`
D. none of these

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the given information
We are given that \( xyz = m \) and the determinant \( p = |(x, y, z), (z, x, y), (y, z, x)| \) is an orthogonal matrix. We also know that \( y = x = z \).

### Step 2: Set up the determinant matrix
The matrix \( p \) can be expressed as:
\[ p = \begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix} \]

### Step 3: Transpose the matrix
The transpose of matrix \( p \) is:
\[ p^T = \begin{pmatrix} x & z & y \\ y & x & z \\ z & y & x \end{pmatrix} \]

### Step 4: Multiply \( p \) by \( p^T \)
We need to compute the product \( p \cdot p^T \):
\[ p \cdot p^T = \begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x \end{pmatrix} \cdot \begin{pmatrix} x & z & y \\ y & x & z \\ z & y & x \end{pmatrix} \]

### Step 5: Calculate the elements of the product
Calculating the first element (1,1):
\[ x^2 + y^2 + z^2 \]
Calculating the first element (1,2):
\[ xz + yx + zy \]
Calculating the first element (1,3):
\[ xy + yz + zx \]
Calculating the second row similarly will give us:
\[ \begin{pmatrix} x^2 + y^2 + z^2 & xz + yx + zy & xy + yz + zx \\ \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots \end{pmatrix} \]

### Step 6: Set the product equal to the identity matrix
Since \( p \) is an orthogonal matrix, we have:
\[ p \cdot p^T = I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 7: Equate the corresponding elements
From the (1,1) position, we get:
\[ x^2 + y^2 + z^2 = 1 \]

### Step 8: Substitute \( y = x = z \)
Let \( x = z = y = k \). Then:
\[ 3k^2 = 1 \]

### Step 9: Solve for \( k \)
\[ k^2 = \frac{1}{3} \implies k = \pm \frac{1}{\sqrt{3}} \]

### Step 10: Conclusion
Thus, \( z = k = \pm \frac{1}{\sqrt{3}} \).

### Final Answer
The value of \( z \) is \( \pm \frac{1}{\sqrt{3}} \).

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