ISI-2004-30

Question

Let $A$ be the matrix $\begin{pmatrix} x&0 &3 \\ -3&y &y \\ 0&0 &1 \end{pmatrix}$. If the determinant of $A^n$ is equal to the determinant of $A$ for all $n \geq 2$, then the locus of the points $(x, y)$ with $xy \neq 0$ is

• A. a parabola
• B. an ellipse
• C. a hyperbola
• D. none of the above.

Comments

Considering n=2

$A^{2}=  \left( \begin{array}{ccc}
x & 0 & 3\\
-3 & y & y\\0 & 0 & 1
\end{array} \right)
%
\left( \begin{array}{ccc}
x & 0 & 3\\
-3 & y & y\\0 & 0 & 1
\end{array} \right) $

$
A^{2}=
  \begin{vmatrix}
   x^{2} & 0 & 3x+3\\
   -3x-3y & y^{2} & -9+y^{2}+y\\
   0 & 0 & 1
  \end{vmatrix} = x^{2}y^{2}$

Determinant of matrix A is xy.
Given Det of $A^{n}$ = Det of A 

$\Rightarrow$ $x^{n}y^{n}= xy$         ( $\because xy\neq0$, dividing by xy)

$\therefore$ xy = 1 is the equation of Hyberbola

Answer : C