If A=cos⁡θ−sin⁡θsin⁡θcos⁡θ, then

We haveAA′=cos⁡θ−sin⁡θsin⁡θcos⁡θcos⁡θsin⁡θ−sin⁡θcos⁡θ=cos2⁡θ+sin2⁡θcos⁡θsin⁡θ−sin⁡θcos⁡θcos⁡θsin⁡θ−sin⁡θcos⁡θsin2⁡θ+cos2⁡θ=1    00    1=I2Similarly, A′A=I2Thus, A is an orthogonal matrix.