Question

If A = `[(costheta, sintheta),(-sintheta, costheta)]` prove that AA^T = I

Answer 1

A^T = `[(costheta, - sintheta),(-sintheta, costheta)]`

A · A^T = `[(costheta, sintheta),(- sintheta, costheta)] [(costheta, - sintheta),(sintheta, costheta)]`

= `[(cos^2theta + sin^2theta, -costhetasintheta + costhetasintheta),(-sintheta costheta + costheta sintheta, sin^2theta + cos^2theta)]`

= `[(1, 0),(0, 1)]`

= I

Hence it is proved.