Question

If P is a 3×3 orthogonal matrix α,β,γ are the angles made by a straight line OX, OY, OZ and A=⎡⎢⎣sin2αsinαsinβsinαsinγsinαsinβsin2βsinβsinγsinαsinγsinβsinγsin2γ⎤⎥⎦&Q=PTAP.

If PQ6PT=KA then k is

Answer 1

$co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$

$\Rightarrow si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma =2$

hence $A^2=2A\Rightarrow A^3=2^{2}A\Rightarrow A^6=2^{5}A$

$A^2=Q=P^{T}AP.P^T{A}^{T}P$
$P.P^T=I$

therefore $A^2=P^T{A}^{2}P$

Similarly $Q^6=P^T{A}^{6}P$

So $P{Q}^6{P}^T=P\left(P^T{A}^{6}P\right)PT$

$=A^6=32A$