# Linear algebra proof

1. Apr 26, 2010

### blackblanx

1. The problem statement, all variables and given/known data

Let A be a 2x2 symmetric matrix and x be a scalar. Prove that the graph of the quadratic equation (X^T)AX = k is hyperbola if k is non zero and det(A) less than zero

(T stands for transpose

2. Relevant equations

not were to begin

2. Apr 27, 2010

### HallsofIvy

Staff Emeritus
Well, begin by writing it out!
Let X= <x, y> and
$$A= \begin{bmatrix} a & b \\ c & d\end{bmatrix}$$.

Then
$$X^TAX= \begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}=ax^2+ cxy+ bxy+ dy^2= ax^2+ (b+ c)xy+ dy^2$$

Now, under what conditions on a, b, c, d is $ax^2+ (b+ c)xy+ dy^2$ a hyperbola?