Consider the quadratic form Q(x, y, z) associated to the matrix

B = \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -2 \end{array}\right]\)

Let

S = \(\left\{\left[\begin{array}{l} \rm a \\\rm b \\\rm c\end{array}\right] \in \rm ℝ^3 \mid Q(a, b, c)=0\right\}\).

Which of the following statements is FALSE?

Concept:

A quadratic form is degenerate if at least one eigenvalue is 0 

Explanation:

B = \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -2 \end{array}\right]\) and S = \(\left\{\left[\begin{array}{l} \rm a \\\rm b \\\rm c\end{array}\right] \in \rm ℝ^3 \mid Q(a, b, c)=0\right\}\).

So quadratic form is Q(x, y, z) = x² + y² -2z² + 2xy

Now, Q(a, b, c) = 0

⇒ a2 + b2 -2c2 + 2ab = 0....(i)

(1): On xy-plane, z = 0 so c = 0

Therefore (i) implies

a 2 + b2 + 2ab = 0 ⇒ (a + b)² = 0 ⇒ a + b = 0

i.e., x + y = 0, which is a line

So option (1) is TRUE

(2) On xz-plane, y = 0 so b = 0

Therefore (i) implies

a2 - 2c2 = 0 ⇒ x2 - 2z2 = 0 which is not an equation of ellipse

So option (2) is FALSE

(3): (i) ⇒  a2 + b2 -2c2 + 2ab = 0

  ⇒ (a + b)² = 2c²

  ⇒ a + b = ± √2c

So S is the union of two planes.

Option (3) is TRUE

(4): B = \(\left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -2 \end{array}\right]\) 

eigenvalues of B are -2, 2, 0

So Q is a degenerate quadratic form.

Option (4) is TRUE