Let and be the distinct roots of the equation x^2+x-1=0. Consider the set T=\ 1, , \ . For a 3 3 matrix M= (a_ i j )_ 3 3 , define R_i=a_ i 1 +a_ i 2 +a_ i 3 and C_j=a_ 1 j +a_ 2 j +a_ 3 j for i=1,2,3 and j=1,2,3. Match each entry in List-I to the correct entry in List-II. The correct option is
C. ( P ) (2) ( Q ) (4) ( R ) (3) ( S ) (5)
aligned & , are roots of x ^2+ x -1=0 \\ & + =-1 1+ + =0 \\ & M= [ array lll a _ 11 & a _ 12 & a _ 13 \\ a _ 21 & a _ 22 & a _ 23 \\ a _ 31 & a _ 32 & a _ 33 array ] aligned (P) M= [ array lll 1 & & \\ & & 1 \\ & 1 & array ] 3! 2=12 For one arrangement of row 1 we can arrange other two rows exactly in two ways and row 1 can be arranged in 3 ! ways 3! 2=12 ways (Q) M= [ array lll x & a & b \\ a & y & c \\ b & c & z array ] Consider one such arrangement with a= , b= , c=1 M= [ array lll 1 & & \\ & & 1 \\ & 1 & array ] a, b, c can be arranged in 3 ! ways and corresponding entries can be arranged in 1 way. (R) [ array ccc 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 array ] [ array l x \\ y \\ z array ]= [ array c a \\ 0 \\ -c array ] aligned & a y+b z=a \\& -a x+c z=0 \\& -b x-c y=-c aligned It is observed that D = D _ x = D _ y = D _ z =0 infinite solution (S) [ array lll 1 & & \\ & & 1 \\ & 1 & array ] -1- ^2+ ^2+ ^2- ^2 =0 ( since = + =-1 )
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