Question

If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, show that A^TA = I, where I is the unit matrix of order 2

Answer 1

A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`

∴ A^T = `[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`

∴ A^TA = `[(cos alpha, -sin alpha),(sin alpha, cos alpha)] [(cos alpha, sin alpha),(-sin alpha, cos alpha)]`

`=[(cos^2alpha + sin^2alpha, sinalpha cos alpha - sin  alpha cosalpha),(sin alpha cos alpha - sin alpha cos alpha, sin^2alpha + cos^2alpha)]`

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

= I, where I is the unit matrix of order 2.