A and B are two 3 3 non-singular matrices such that adj A=|A| B . If tr (x) denotes the trace of a square matrix X and C= [ array ccc 4 & 4 & 7 3 & -2 & 5 -2 & 3 & 6 array ] , then _ k=1 ^x tr ( 1 3^k (A B)^k C ) is equal to

Step-by-step solution

aligned & Given, adj A=|A| B & B= adj A |A| =A⁻¹ & and tr (C)=4-2+6=8 & and (A B)^k= (A A⁻¹ )^k=(I)^k=I & Now, _ k=1 ^ tr ( 1 3^k (A B)^k C )= _ k=1 ^ tr ( 1 3^k I C ) & = _ k=1 ^ tr ( 1 3^k C )= tr (C) _ k=1 ^ 1 3^k =8 [ 1 3 + 1 3^2 + 1 3^3 + ] & =8 [ 1 3 1- 1 3 ]=8 [ 1 3 3 3 ] & =8 1 3 3 2 =4 & aligned