Question

Answer the following question:

If A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]` and A + A^T = I, where I is unit matrix 2 × 2, then find the value of α

Answer 1

A = `[(cosalpha, -sinalpha),(sinalpha, cosalpha)]`

∴ A^T = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`

∴ A + A^T = I gives

`[(cosalpha, -sinalpha),(sinalpha, cosalpha)] + [(cosalpha, sinalpha),(-sinalpha, cosalpha)] = [(1, 0),(0, 1)]`

∴ `[(2cosalpha, 0),(0, 2cosalpha)] = [(1, 0),(0, 1)]`

∴ by equality of matrices,

2cos α = 1

∴ cos α = `1/2`

∴ cos α = `cos  pi/3`

∴ α = `pi/3 or 60^circ`