Let A=[[|gathered 0 - 0 1 0 0 gathered]]|.

Question

Let $A=[[|\begin{gathered} \cos 	heta  \quad  0 \quad  -\sin 	heta \ 0 \quad  1 \quad  0 \ \sin 	heta  \quad  0 \quad  \cos 	heta \end{gathered}]]|$. If for some $	heta \in (0,\pi ),{A}^{2}={A}^{T}$, then the sum of the diagonal elements of the matrix $(A+I{)}^{3}+(A-I{)}^{3}-6A$ is equal to \_\_\_\_\_\_\_\_\_ .

Answer: 6

Accepted Answer

Note that $A$ is orthogonal:

$A{A}^{T}={A}^{T}A=I{A}^{T}={A}^{-1}$

Given ${A}^{2}={A}^{T}$, then:

$\begin{gathered} {A}^{3}=I \ Tr(A+I{)}^{3}+(A-l{)}^{3}-6A=Tr(2{A}^{3}+6A-6A) \ =Tr(2{A}^{3})=Tr(2I) \ ((A+l{)}^{3}+(A-I{)}^{3}=2{A}^{3}+6A2{A}^{3}=2I)= \ 6 \end{gathered}$

Solutions · 1