Let P_1=[ 1 0 0; 0 1 0; 0 0 1 ], P_2=[ 1 0 0; 0 0 1; 0 1 0 ], P_3=[ 0 1 0; 1 1 0; 0 0 1 ], P_4=[ 0 1 0; 0 0 1; 1 0 1 ], P_5=[ 0 0 1; 1 0 0; 0 1 0 ], P_6=[ 0 0 1; 0 1 0; 1 0 0 ], and X = Σ_(k=1)^6 P_k [ 2 1 3; 1 0 2; 3 2 1 ] P_k^T. Where P_k^T denotes the transpose of matrix P_k. Then which of the following option is/are correct?

Correct option is C. If X[ 1 1 1 ]=α[ 1 1 1 ] then α=30ClearlyP_1=P_1^T=P_1^-1P_2=P_2^T=P_2^-1...P_6=P_6^T=P_6^-1and A^T=A, Where A = [ 2 1 3; 1 0 2; 3 2 1 ]Using Formula (A+B)^T=A^T+B^TX^T=(P_1AP_1^T+...+P_6AP_6^T)^T=P_1A^TP_1^T+.+P_6A^TP_6^T=X ⇒ X is symmetricLet B= [ 1 1 1 ]XB=P_1AP_1^TB+P_2AP_2^T+.+P_6AP_6^T B=P_1AB+PAB+.+P_6ABXB=(P_1+P_2+..+P_6) [ 1 1 1 ][ 6× 2+3× +6× 2 6× 2+3× 2+6× 2 6× 2+3× 2+6× 2 ]=[ 30 30 30 ]=30 B⇒α=30Since X[ 1 1 1 ]=30 [ 1 1 1 ]⇒ (X-30I)B=0 has a non trivial solution B=[ 1 1 1 ]⇒ |x-30 I|0X=P_1AP_1^T+...+P_6AP_6^Ttrance (x)=t_r(P_1AP_1^T)+..+t_r(P_6AP_6^T)=(2+0+1)+..+(2+0+1)=3× 6=18