Let A= [ array ccc & 0 & - 0 & 1 & 0 & 0 & array ] . If for some (0, ) , A^2=A^T , then the sum of the diagonal elements of the matrix ( A + I )^3+( A - I )^3-6 ~A is equal to _____ .

Step-by-step solution

A is orthogonal matrix aligned & A^T=A⁻¹ & A^2=A⁻¹ ( A ^2= A ^ T ) & A^3=I & let B=(A+I)^3+(A-I)^3-6 A & =2 (A^3+3 A )-6 A & =2 A^3 & B=2 I= [ array lll 2 & 0 & 0 0 & 2 & 0 0 & 0 & 2 array ] aligned Now sum of diagonal elements =2+2+2=6