if for some non-zero vector then which means -1 is an eigenvalue of contradiction! so is non-singular. to show that S is skew symmetric:

thus: but: thus: i.e. isii)If B is a skew-symmetric matrix ,then C = (I – B)( I+B)-1 is an orthogonal matrix with no eigenvalue equal to –1.

orthogonal. to show that has no eigenvalue equal to -1 let then: hence is non-singular and so has no eigenvalue equal to -1.