Let R = ( x 0 0 0 y 0 0 0 z ) be a non-zero 3 × 3 matrix, where x sin = y sin = z sin. For a square matrix M , let trace ( M ) denote the sum of all the diagonal entries of M . Then, among the statements:

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Solution

Q.
Let $R = (\begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix})$ be a non-zero $3 \times 3$ matrix, where $x \sin θ = y \sin (θ + \frac{2 π}{3}) = z \sin (θ + \frac{4 π}{3}) \neq 0, θ \in (0, 2 π) .$ 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 trace $(adj(adj (R))) = 0$ , then $R$ has exactly one non-zero entry.

1 Only (I) is true.

2 Both (I) and (II) are true.

3 Neither (I) nor (II) is true.

4 Only (II) is true.

Ans.
(4)

We have, $x \sin θ = y \sin (θ + \frac{2 π}{3}) = z \sin (θ + \frac{4 π}{3})$

$\Rightarrow y = \frac{x \sin θ}{\sin (θ + \frac{2 π}{3})}$ and $z = \frac{x \sin θ}{\sin (θ + \frac{4 π}{3})}$

$∴$ $x + y + z = x + \frac{x \sin θ}{\sin (θ + \frac{2 π}{3})} + \frac{x \sin θ}{\sin (θ + \frac{4 π}{3})}$ $= \frac{- 3 x}{4 \sin (θ + \frac{2 π}{3}) \sin (θ + \frac{4 π}{3})} \neq 0$

Also, $R = (\begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix}) or adj R = (\begin{matrix} y z & 0 & 0 \\ 0 & x z & 0 \\ 0 & 0 & x y \end{matrix})$

$\Rightarrow adj(adj R)$ $= (\begin{matrix} x^{2} y z & 0 & 0 \\ 0 & y^{2} x z & 0 \\ 0 & 0 & z^{2} x y \end{matrix})$

$∴$ $Tr(adj(adj R)) = x y z (x + y + z) \neq 0$

Only (II) is true.

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