If

 \(A = \begin{bmatrix} y & z & x \\ z & x & y \\ x & y & z \end{bmatrix} \)

where x,y,z are integers, is an orthogonal matrix, then what is the value of $x{}^2+y{}^2+z{}^2$?

Calculation:

Given,

The matrix A is:

\( A = \begin{bmatrix} y & x & x \\ z & x & y \\ x & y & z \end{bmatrix} \)

Since A  is an orthogonal matrix, we know that:

\( A^T = A^{-1} \quad \Rightarrow \quad A^T A = I \)

This property tells us that A  is orthogonal, and it implies that \(A^T A \) (the product of A's transpose and A is equal to the identity matrix I , which is:

\( A^T A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)

Now, let’s calculate \(A^T A \) step by step. The transpose of matrix A , denoted \(A^T \) is:

\( A^T = \begin{bmatrix} y & z & x \\ x & x & y \\ x & y & z \end{bmatrix} \)

Now, we perform matrix multiplication between \(A^T\) and A:

\( A^T A = \begin{bmatrix} y & z & x \\ x & x & y \\ x & y & z \end{bmatrix} \begin{bmatrix} y & x & x \\ z & x & y \\ x & y & z \end{bmatrix} \)

Performing this multiplication, we get the following matrix:

\( A^T A = \begin{bmatrix} y^2 + z^2 + x^2 & xy + zx + xy & xz + yz + x^2 \\ xy + zx + xy & x^2 + x^2 + y^2 & xy + xz + yz \\ xz + yz + x^2 & xy + xz + yz & x^2 + y^2 + z^2 \end{bmatrix} \)

This matrix must be equal to the identity matrix I , which is:

\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)

By comparing the elements of the matrices, we get the following system of equations:

Thus, the key result from the orthogonality condition is:

\( x^2 + y^2 + z^2 = 1 \)

Hence, the correct answer is Option 2.