Let R=⎛⎝x000y000z⎞⎠ be a non-zero 3×3 matrix, where xsinθ=ysin(θ+2π3)=zsin(θ+4π3) ≠0,θ∈(0,2π). 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 trace (adj(adj(R))=0, then R has exactly one non-zero entry.
Question
Let R=⎛⎝x000y000z⎞⎠ be a non-zero 3×3 matrix, where xsinθ=ysin(θ+2π3)=zsin(θ+4π3) ≠0,θ∈(0,2π). 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 trace (adj(adj(R))=0, then R has exactly one non-zero entry.
Solution
The given matrix R is a diagonal matrix with entries x, y, and z on the diagonal. The trace of a matrix is the sum of its diagonal entries. Therefore, the trace of R is x + y + z.
Given that xsinθ = ysin(θ+2π/3) = zsin(θ+4π/3), we can say that x, y, and z are all equal to some common value (let's call it a) times different sine values. Therefore, x = asinθ, y = asin(θ+2π/3), and z = asin(θ+4π/3).
Adding these together, the trace of R becomes asinθ + asin(θ+2π/3) + asin(θ+4π/3). This is a sum of sine functions with the same amplitude but different phases. The sum of these sine functions is not necessarily zero, so statement (I) is not always true.
The adjugate of a matrix is the transpose of its cofactor matrix. The adjugate of the adjugate of a matrix (adj(adj(R))) is a matrix that depends on the entries of R. If the trace of this matrix is zero, it does not necessarily mean that R has exactly one non-zero entry. Therefore, statement (II) is also not always true.
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