Question

If A =  `([cos alpha, sin alpha],[-sinalpha, cos alpha])` , find α satisfying 0 < α < `pi/r`when `A+A^T=sqrt2I_2` where A^T is transpose of A.

Answer 1

Consider the given matrix

A = `[(cosalpha, sin alpha),(-sinalpha,cos alpha)]0<alpha<pi/2`

`A+A^T=sqrt2I_2`

`[(cosalpha, sin alpha),(-sinalpha, cos alpha)]+[(cosalpha,-sinalpha),(sinalpha,cos alpha)]=sqrt2[(1,0),(0,1)]`

`[(2cosalpha, 0),(0,2cosalpha)]=[(sqrt2,0),(0, sqrt2)]`

`2cosalpha=sqrt2`

`cosalpha=sqrt2/2=1/sqrt2`

`alpha=pi/4`