Let A = \ X = (x, y, z) ^ T : PX = 0and x ^ 2 + y ^ 2 + z ^ 2 = 1\...

$∵ \left|P\right| = 0$

$∴$ System of equations

$x + 2y + z = 0$

$2x - 3y + 4z = 0$

$x + 9y - z = 0$ has infinitely many solutions.

Let $z = k \in R$ then $x = - \dfrac{11k}{7}, y = \dfrac{2k}{7}, z = k$

Also ${x}^{2} + {y}^{2} + {z}^{2} = 1$

$∴ k = \pm \dfrac{7}{\sqrt{174}}$

$∴$ Two solutions are only possible.

Hence, set $A$ contains exactly two elements.