<prompt> Acha mathbfD iwe matrix inayowakilisha upanuzi wenye kipimo cha k 0, na mathbfR iwe matrix inayowakilisha mzunguko kuhusu asili kwa pembe ya theta counter -saa. KamamathbfD mathbfR anzapmatrix -7 1 -1 -7 mwishopmatrix,kisha tafuta k.  <response>: Tunayo mathbfD anzapmatrix k 0 0 k endpmatrix na mathbfR anzapmatrix cos theta -sin theta sin theta cos theta mwishopmatrix, hivyomathbfD mathbfR anzapmatrix k 0 0 k mwishopmatrix anzapmatrix cos theta -sin theta dhambi theta cos theta mwishopmatrix anzapmatrix k cos theta -k sin theta k sin theta k cos theta mwishopmatrix. Kwa hivyo, k cos theta -7 na k sin theta -1. Kishak2 cos2 theta k2 sin2 theta 49 1 50,ambayo hurahisisha k2 50. Tangu k 0, k sqrt50 5 sqrt2. 