Let R={( able x , 0 , 0 ; 0 , y , 0 ; 0 , 0 , z)} be a non-zero 3 × 3 matrix, where {\x} sin \;θ= {\y} sin \; (θ+\;{2 π}/{3}) = {\z} sin \; (θ+\;{4 π}/{3}) ≠ 0, θ ∈(0,2 π). For a square matrix M, let trace (M) denote the sum of all the diagonal entries of M. Then, among the statements:(I) Trace ( {\R})=0(II) If { race}({\adj}({\adj}( {\R}))=0, then {\R} has exactly one non-zero entry. [30-Jan-2024 Shift 2]

Question

Let R={(	able x , 0 , 0 ; 0 , y , 0 ; 0 , 0 , z)} be a non-zero 3 × 3 matrix, where  {\x}  sin \;θ= {\y}  sin \; (θ+\;{2 π}/{3}) = {\z}  sin \; (θ+\;{4 π}/{3})  ≠ 0, θ ∈(0,2 π). For a square matrix M, let trace (M) denote the sum of all the diagonal entries of M. Then, among the statements:(I) Trace ( {\R})=0(II) If {	race}({\adj}({\adj}( {\R}))=0, then  {\R} has exactly one non-zero entry.  [30-Jan-2024 Shift 2]

Examsnet · Accepted Answer

Correct Answer is :  Only (II) is true

JEE Main 30-Jan-2024 Shift 2 Solved Paper