91Ó°ÊÓ

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors in linear algebra.

Where ${\mathrm{Q}}^{\mathrm{T}}$is the transpose of Q and l is the identity matrix is one approach to describe this.

Since A has three distinct eigenvalues, then we have possible distinct orthonormal eigenvectors corresponding to these eigenvalues:

$Â\pm {\stackrel{â‡\P Ä}{\mathrm{v}}}_1,Â\pm {\stackrel{â‡\P Ä}{\mathrm{v}}}_2,\mathrm{and}Â\pm {\stackrel{â‡\P Ä}{\mathrm{v}}}_3$

Now, when obtaining the orthogonal matrix S, the first column would have 6 choices, moving to the second column we will have 4 choices (since we can't have a corresponding eigenvector to the same eigenvalue used in the first column), and finally moving to the third column we will have 2 choices left.

Therefore 6x4x2=48

The different orthogonal matrices are 48.