Show that there does not exist any 4 × 4 real symmetric orthogonal matrix with all diagonal elements zero

Question : Show that there does not exist any 4 × 4 real symmetric orthogonal matrix with all diagonal elements zero

Solution :

Let A be a 4 × 4 real symmetric orthogonal matrix.

\[=> A^T= A\]

and

\[ AA^T=A^TA=I\]

from these two equations

\[=>A^2=I\]

\[=> (A-I)(A+I)=0\]

\[.=>A=I or A= -I\]

Therefore all of diagonal entries of A must be either 1 or -1.

Hence it is not possible for a matrix to be symmetric orthogonal with all diagonal elements zero.